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PRACTICAL 


COMMERCIAL  ARITHMETIC 


DESIGNED    FOR    USE    IN    ALL   SCHOOLS    IX    WHICH    THE 


COMMERCIAL  BRANCHES  ARE  TAUGHT 


AND   AS   A 


BOOK   OF   REFEREJSTCE 


*  FOR 


BUSIJ^ESS   MEiSr. 


1889. 


Entered  according  to  Act  of  Congress,  in  tlie  year  1888, 

By  WILLIAMS  &  ROGEES, 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington,  D.  C. 


E.  R.  ANDREWS,  PRINTER  AND  BOOKBINDER 
RIXIH ESTER,  N.  Y. 


SRUR 


^^/^^^T^^^^ 


PREFACE 


TVrO  school  text-book  is  used  now-a-days  in  consequence  of  prefatory  compli- 
^  ments  paid  it  by  its  authors  or  publishers.  In  this  day  of  general  enlight- 
enment, teachers  understand  the  necessities  of  their  classes,  and,  as  a  rule,  need 
no  advice  as  to  what  or  how  to  teach. 

Since  the  wants  of  American  schools  and  the  ideas  of  American  teachers  are 
various,  a  variety  of  text-books  upon  every  topic  of  school  instruction  is 
required,  and  with  the  hope  and  belief  that  .the  contents  of  this  volume  will 
more  nearly  meet  the  necessities  of  some  schools  and  the  ideas  of  some  teachers, 
than  any  of  the  several  good  books  upon  the  subject  of  Arithmetic  now  in  print, 
the  work  is  respectfully  submitted  by 

THE   AUTHORS. 


CONTENTS. 


SIMPLE    NUMBERS. 


l-AGK 

Definitions 1 

Signs 3 

Abbreviations  and  Contractions 3 

Notation  and  Numeration -.  3 

Arabic  Metliod  of  Notation 4 

Frencli  System  of  Numeration 4 

English  System  of  Numeration. 5 

Roman  Notation  and  Numeration 7 

Addition 8 

Addition  Table. - 9 

Group  Method  of  Addition 11 

Horizontal  Addition 11 


Subtraction 16 

Subtraction  Table 17 

Multiplication *. 20 

Multiplication  Table 21 

Division 28 

Long  Division 34 

Average 37 

Complement 37 

Factors  and  Factoring 38 

Divisors 39 

Multiples 41 

Cancellation 43 


COMMON    FRACTIONS. 


Definitions 45 

Reduction  of  Fractions 46 

Addition  of  Fractions 49 

Subtraction  of  Fractions 51 


Multiplication  of  Fractions 54 

Division  of  Fractions 57 

Complex  Fractions 61 

Miscellaneous  Examples 61 


DECIMALS. 


Definitions 66 

Numeration  of  Decimals .-  67 

Notation  of  Decimals 68 

Reduction  of  Decimals  .-. 69 

Circulating  Decimals 72 


Addition  of  Decimals 73 

Subtraction  of  Decimals 75 

Multiplication  of  Decimals 75 

Division  of  Decimals 76 

Miscellaneous  Examples 80 


UNITED    STATES    MONEY 


Definitions 

United  States  Coins 

United  States  Paper  ]\Ioney 

Reduction  of  United  States  Money 


81 
82 
83 
84 


Addition  of  United  States  Monej' 84 

Subtraction  of  United  States  Money 84 

Multiplication  of  United  States  Money..  85 

Division  of  United  States  Money  _ 86 


Definitions . 


ANALYSIS. 
87     I     Examjjles . 


87 


SPECIAL  APPLICATIONS. 


Definitions 89 

Aliquot  Parts 89 

Instructions  for  Pr^rtice  with  Ali»iuot 

Parts DO 

Miscellaneous  Contractions 90 


Instructions  for  Finding  Quantity 91 

Miscellaneous  Contractions 92 

Bills,  Statements,  and  Inventories 100 

Miscellaneous  E.xamples 106 


CONTENTS. 


Vll 


DENOMINATE    NUMBERS 


PAGE 

Definitions 108 

Measures  of  Time 108 

Reduction  of  Time. 110 

Addition  of  Time 111 

Subtraction  of  Time -  112 

Circular  Measure 113 

Latitude,  Longitude,  and  Time 113 

Standard  Time 114 

English  Money 116 

Reduction  of  English  Money 117 

Measures  of  Weight 119 

Troy  Weight^ 119 

Reduction  of  Denominate  Numbers  ...  119 

Addition  of  Denominate  Numbers 122 

Subtraction  of  Denominate  Numbers  ..  123 
Multiplication  of  Denominate  Numbers  123 

Division  of  Denominate  Numbers 123 

Avoirdupois  Weight 125 

Table  of  Avoirdupois  Pounds  per  Bushel  126 

Additional  Tables 127 

Apothecaries'  Weight. 128 

Comparative  Table  of  Weights 128 

Measures  of  Capacity. 129 

Dry  Measure 129 


Liquid  Measure 130 

Comparative  Table  of  Liquid  and  Dry 

Measures 130 

Measures  of  Extension 131 

Linear  Measure 131 

Special  Table,  Linear  Measure 132 

Square  Pleasure  . 132 

Involution 137 

Evolution ^ 137 

Square  Root 137 

Surveyors'  Long  Measure 141 

Surveyors'  Square  Measure 142 

Cubic  Measure ^. 143 

Table  Special  Cubic  Measures 144 

Produces'  and   Dealers'  Approximate 

Rules 146 

Hay  Measurements 147 

Cube  Root 147 

Duodecimals 151 

Miscellaneous  Measurements 151 

The  Metric  System 155 

French  3Ioney 157 

German  Money 157 

M  iscellaneous  Examples 158 


PERCENTAGE. 


Definitions 160 

To  find  the  Percentage,  the  Base  and 

Rate  being  given 162 

To  find  the  Base,  the  Percentage  and 

Rate  being  given  ._ 163 

To  find  the  Rate,  the  Percentage  and 

Base  being  given 164 

To  find   the   Amount   Per    Cent.,    the 

Rate  being  given 165 

To  find  the  Difference   Per  Cent.,  the 

Rate  being  given 166 

To  find  the  Amount,  the  Base  and  Rate 

being  given 166 

To  find  the  Difference,   the  Base  and 

Rate  being  given _.  167 

To  find  the  Base,  the  Amount  or  Differ- 
ence, and  the  Rate  being  given 168 

Review  of  the  Principles  of  Percentage  169 

Profit  and  Loss 173 

To  find  the  Profit  or  Loss,  the  Cost  and 

Rate  being  given 173 

To  find  the  Cost,  the  Gain  or  Loss,  and 

the  Rate  of  Gain,  or  Loss  being  given  174 


To  find  the  Rate  of  Profit  or  Loss,  the 
Cost  and  the  Profit  or  Loss  being 
given - 176 

To  find  the  Cost,  the  Selling  Price  and 
the  Rate  Per  Cent,  of  Profit  or  Loss 
being  given 177 

Review  of  the  Principles  of  Profit  and 
Loss 178 

Tkade  Dtscouxt 183 

To  find  the  Selling  Price,  the  List  Price 

and  Discount  Series  being  given 183 

To  find  the  Price  at  which  Goods  must 
lie  Marked  to  Insure  a  Given  Per 
Cent,  of  Profit  or  Loss,  the  Cost  and 
Discount  Series  being  given 184 

To  find  a  Simple  Equivalent  Per  Cent, 
of  Discount,  a  Discount  Series  being 
given. , 186 

Storage..   '. 187 

To  find  the  Simple  Average  Storage 187 

To  find  the  Charge  for  Storage  with 
Credits 188 

To  find  the  Storage  when  Charges  Vary  190 


Tin 


COXTENTS. 


TAGS 

Commission 191 

To  find  the  Commission,  the  Cost  or  Sell- 
ing Price  and  Per  Cent,  of  Commis- 
sion being  given 192 

To  find  tlie  Investment  or  Gross  Sales, 
the  Commission  and  Per  Cent,  of 

Commission  being  given 192 

To  find  the  Investment  and  Commission, 
when  both  are  included  in  a  Remit- 
tance by  the  Principal 192 

CcsTOM-HorsE  BrsixEss 197 

To  find  Specific  Duty.. 196 

To  find  .Vd  Valorem  Duty 199 

Taxes 201 

To  find  Property  Tax 201 

To  find  a  General  Tax 202 

Insurance. = 204 

To  find  the  Cost  of  Insurance 206 

To  find  the  Amount  Insured,  the  Pr<>- 
mium  and  Per   Cent,  of  Premium 

being  given 206 

Peuso5>  AL  Insurance 208 

Interest 209 

Six  Per  Cent.  Method 210 

To  find  the  Interest  on  Any  Sum  of 
Money,  at  Other  Rates  than  6  Per 

Cent 210 

To  find  the  Interest,  the  Principal,  Rate, 

and  Time  being  given 211 

To  find  the  Principal,  the  Interest,  Rate, 

and  Time  being  given 214 

To  find  the  Principal,  the  Amount,  Rate, 

and  Time  being  given 21 1 

To  find  the  Rate,  the  Principal,  Interest, 

and  Time  being  given. 215 

To  find  the  Time,  the  Principal,  Interest, 

and  Rate  being  given. 216 

SnoKT  Methods  kou  Finding  Interest  216 
To  find  Interest  for  Days,  at  6  Per  Cent., 

360  Day  Basis.. 217 


To  find  Interest  at  Other  Rates  than  6 

Per  Cent.,  360  Day  Basis 217 

To  find  Interest  for  Days  at  6  Per  Cent., 

365  Day  Basis 220 

Periodic  Interest 221 

To  find  Periodic  Interest 221 

Compound  Interest 223 

To  lind  Compound  Interes' 223 

Compound  Interest  Table 224 

True  Discount 230 

To  find  the  Present  Worth  of  a  Debt...  230 

Bank  Discount 233 

General  Remarks  on  Commercial  Paper,  234 
To  find  the  Discount  and  Proceeds  of  a 

Note 236 

To  find  the  Face  of  a  Note 238 

Partial  Paysients 239 

United  States  Rule 240 

Merchants'  Rule .•  241 

Equation  of  Accounts 243 

"When  the  Items  are  all  Debits,  or  all 
Credits,    and    Lave    no    Terms    of 

Credit 244 

"When  the  Items  have  Different  Dates 
and  the  Same  or  Different  Terms  of 

Credit 248 

"When  an  Account  has  Both  Debits  and 

Credits 250 

R.\.Tio 259 

Proportion 260 

Simple  Proportion .. 260 

Compound  Proportion 261 

Partnership 263 

To  Divide  the  Gain  or  Loss,  when  Each 
Partner's  Investment  has  been  Em- 
ployed for  the  Same  Period  of  Time.  264 
To  Divide  the  Gain  or  Loss,  According 
to  the  Amount  of  Capital  Invested 

and  the  Time  it  is  Employed 266 

Answers 273 


COMMERCIAL    ARITHMETIC. 


DEFINITIONS. 

1.  Arithmetic  is  the  Science  of  Numbers  and  the  Art  of  Computation. 

2.  A  Unit  is  a  single  thing, 

3.  A  Number  is  a  unit  or  a  collection  of  units. 

4.  The  Unit  of  a  number  is  one  of  the  collection  of  units  forming  the 
."number;  thus,  the  unit  of  5  is  1;  of  IT  dollars,  1  dollar;  of  30  pupils,  1  juipil. 

5.  An  Integer  is  a  whole  or  entire  number. 

6.  An  Even  Number  is  one  that  can  be  exactly  divided  by  2;  as,  6,  8,  44. 

7.  An  Odd  Number  is  one  that  cannot  be  exactly  divided  by  2;  as,  5,"  9,  23. 

8.  A  Composite  Number  is  one  tluit  can  be  resolved  or  separated  into 
factors;  as,  4  =  2  X  2;  12  =  3  X  2  x  2. 

9.  A  Prime  Number  is  one  that  cannot  be  resolved  or  separated  into  factors, 
being  divisible  only  by  itself  and  unity;  as,  1,  2,  3,  5,  7,  19,  83. 

10.  An  Abstract  Number  is  one  used  without  reference  to  any  particular 
thing  or  quantity;  as,  3,  11,  24. 

11.  A  Concrete  Number  is  one  used  with  reference  to  some  particular  tiling 
or  quantity;  as,  3  dollars,  11  men,  24  cords  of  wood. 

12.  A  Compound  Denominate,  or  Compound  Number,  is  a  concrete 
number  expressed  by  two  or  more  orders  of  units;  as,  3  dollars  and  11  cents; 
5  pounds,  2  ounces  and  15  pennyweights. 

13.  Like  Numbers  are  such  as  have  the  same  unit  value;  as,  5,  14,  37;  or, 
5  men,  14  men,  37  men;  or,  if  denominate,  the  same  kind  of  quantity;  as,  5 
hours  14  minutes  37  seconds. 

14r.  Unlike  Numbers  are  such  as  have  different  unit  values;  as,  11,  16  days, 
365  dollars,  5  pounds,  4  yards. 

*15.  Ratio  is  the  comparison  of  magnitudes.  It  is  «>f  two  kinds;  urithmetical 
and  geometrical. 

16.  Arithmetical  Ratio  expresses  a  difference. 

17.  Geometrical  Ratio  expresses  a  quotient. 

18.  A  Problem  in  Arithmetic  is  a  question  to  be  solved;  its  analysts,  the 
logical  statement  of  its  conditions  and  of  the  steps  required  for  its  solution. 


2  SIGNS. 

19.  The  Conclusion  of  llio  analysis  is  called  the  (nm^ccr,  or  rcsnU. 

20.  A  Rule  is  an  outline  of  the  etei)s  to  be  taken  in  a  solution. 

SIGNS. 

21.  A  Sign  is  a  character  used  to  express  a  relation  of  terms  or  to  indicate 
an  operation  to  be  performed. 

The  followin2:  are  the  ])rincipal  and  most  useful  arithmetical  signs: 

22.  The  Sign  of  Addition  is  a  perpendicular  cross,  +.  It  is  called  Plus, 
and  indicates  that  the  numbers  betAveen  which  it  is  placed  are  to  be  added  ; 
thus,  5  +  4  indicates  that  4  is  to  be  added  to  5. 

23.  The  Sign  of  Subtraction  is  a  short  horizontal  line,  — .  It  is  called 
Minus,  and  indicates,  when  jjlacod  between  two  numbers,  that  the  value  of  the 
number  on  its  right  is  to  be  taken  from  the  value  of  tlie  number  on  its  left ; 
thus,  8  —  3,  indicates  that  3  is  to  be  subtracted  from  8. 

24.  The  Sign  of  Multiplication  is  an  oblique  cross,  x.  It  indicates  that 
the  numbers  between  which  it  is  placed  are  to  be  multiplied  together;  thus, 
7x9,  indicates  that  the  value  of  7  is  to  be  taken  9  times. 

25.  The  Common  Sign  of  Dirision  is  a  short  horizontal  line  with  a  point 
above  and  one  below,  -=-.  It  indicates  a  comparison  of  numbers  to  determine  a 
quotient,  it  being  understood  that  the  number  at  the  left  of  the  sign  is  to  be  divided 
by  the  one  at  its  right ;  thus.  20  -^  5,  indicates  that  20  is  to  be  divided  by  5. 

26.  The  Sign  of  Ratio  is  the  colon,  :  ;  it  also  indicates  division. 

27.  The  Sign  of  Equality  is  two  short  horizontal  lines,  =.  It  is  read  equals, 
or,  is  eqital  to,  and  indicates  that  the  numbers,  or  expressions,  between  which  it 
is  placed  are  equal  to  each  other;  thus,  2  +  2  =  4, 

28.  The  Signs  of  Aggregation  are  the  parenthesis,  (  ),  brackets,  [  ]' 

brace,  {    },  and  vinculum, .     They  indicate  that  the  quantities  included 

within,  or  connected  by  them,  are  to  be  taken  together  and  su})jected  to  the 
same  operation. 

29.  The  Index,  or  Power  Sign,  is  a  small  figure  placed  at  the  right  of  and 
above  another  figure.  It  indicates  that  the  number  over  which  it  is  placed  is 
to  be  taken  as  a  factor  a  number  of  times  equal  to  the  numerical  value  of  the 
index.  Thus  4^  indicates  that  4  is  to  be  taken  twice  as  a  factor,  or  multiplied 
by  itself  once;  4^  indicates  that  4  is  to  be  used  three  times  as  a  factor.  4-  is  reac^ 
4  squared;  4^  is  read  4  cubed;  also,  the  second  jjower  of  4;  the  third  power  of  4. 

30.  The  Root,  or  Radical  Sign,  is  the  character,  \/',  it  is  the  opposite  of 
the  index,  or  power  sign.  When  there  is  no  figure  in  the  opening,  it  indicates 
that  the  quantity  over  which  the  sign  is  placed  is  to  be  sei)aratcd  into  two  equal 
factors,  or  its  square  root  taken.  A  figure  jdaced  in  the  ojiening  indicates  the 
number  of  equal  factors  required,  or  the  root  to  be  extracted  ;  as,  '\/T\,  \/~~6' 


ABBREVIATIONS   AND   CONTKACTIONS.  3 

31.  The  Dollar  Sign  is  the  character,  %. 

32.  The  Cent  8ig|l  is  the  character,  </: 

33.  The  Decimal  Point  is  the  period,  .  ;  Avhen  employed  to  separate  dollars 
from  cents  it  is  called  a  Separair'ix. 

Fractional  parts  of  a  dollar  are  expressed  only  as  hundredths;  thus  $14.53  is 
read  14  dollars  and  53  cents,  or  14  and  53  hundredths  dollars. 


ABBREVIATIONS  AND  CONTRACTIONS. 

34.  The  following  are  some  of  the  principal  abbreviations  and  contractions  in 
common  use: 


Bbl.  or  Bar.  for  barrel  or  barrels. 

Bu.  for  bushel  or  bushels. 

Cd.  for  cord  or  cords. 

Ct.  for  cent  or  cents. 

Cwt.  for  hundred  weight  or  hundred 

weights. 
Cent,  for  cental  or  centals, 
Da.  for  day  or  days. 
Doz.  for  dozen  or  dozens. 
Ft.  for  foot  or  feet. 


Gal.  for  gallon  or  gallons. 
Hhd.  for  hogshead  or  hogsheads. 
In.  for  inch  or  inches. 
Lb.  for  pound  or  pounds. 
Mo.  for  month  or  months. 
Oz.  for  ounce  or  ounces. 
Pk.  for  peck  or  pecks. 
Pt.  for  pint  or  pints. 
Qt.  for  quart  or  quarts. 
Yd.  for  yard  or  yards. 


Note.  —  Other  and  more  complete  lists  of  abbreviations  and  contractions,  together  with 
illustrations  of  their  uses,  will  be  given  in  the  advanced  part  of  this  work.. 


NOTATION  AND  NUMERATION. 

35.  Notation  is  the  method  of  expressing  numbers. 
There  are  three  ways  of  expressing  numbers. 

I.  By  Words ;  as  one,  two,  three. 

II.  By  Figures,  called,  the  Arabic,  or,  more  properly,  the  Indian  Xotation ; 
this  notation  employs  the  nine  digits,  1,  2,  3,  4,  5,  6,  7,  8,  9,  and  the  naugiit,  0, 
which  is  also  called  zero,  and  cipher. 

By  this  method  a  number  is  written  and  read  with  direct  reference  to  its  successive  periods, 
commencing  with  the  highest. 

III.  By  Letters,  called  the  Romcm  Notation;  this  notation  employs  the  seven 
capital  letters  ;  I  for  one,  V  for  five,  X  for  ten,  L  for  fifty,  C  for  one  hundred, 
D  for  five  hundred,  and  M  for  one  thousand. 

By  this  method  a  number  is  written  and  read  with  direct  reference  to  its  successive  orders, 
and  multiplication  by  one  thousand  is  indicated  by  over-scoring  the  letter  whose  value  is 
to  be  so  increased  ;  as,  V  for  five,  V  for  five  thousand;  31  for  one  tliousand,  M  for  a  thousand 
thousand,  or  a  million. 

36.  Numeration  is  the  method  of  reading  numbers  expressed  by  words, 
figures,  or  letters. 


4  NOTATION    AND    NUMERATION. 

ARABIC  METHOD  OF  NOTATION. 

37.  By  the  Arabic  Method  the  value  of  numbers  increases  from  riglit  to 
left,  and  decreases  from  left  to  right  in  a  ten-fold  ratio;  the  successive  figures 
from  right  to  left  or  from  left  to  right  are  called  orders  of  units,  the  value  of 
one  of  any  order  being  ten  times  the  value  of  one  of  the  order  next  to  its  right, 
and  only  one-tenth  the  value  of  one  of  the  order  next  to  its  left;  for  example,  in 
the  number  one  hundred  and  eleven,  expressed  111,  the  second  1  is  equal  in  value 
to  ten  times  the  first  1,  but  to  only  one-tenth  the  value  of  the  third  1. 

The  succession  of  the  orders  of  units  in  writing  numbers  by  this  method, 
establishes  a  decimal  system  in  whicli  the  numbers  are  divided  for  convenience 
into  periods  of  three  figures,  or  places,  each.  Numbers  so  written  are  read  or 
enumerated  from  right  to  left  to  ascertain  their  value,  and  from  left  to  right  to 
announce  their  value.  The  naught,  or  ci]iher,  is  always  read  as  of  the  order  of 
the  place  it  occupies. 

For  example,  in  reading  to  ascertain  the  value  of  the  expression  265017,  we 
begin  at  the  right  and  name  the  successive  orders  of  units:  units,  tens,  hundreds, 
thousands,  tens  of  thousands,  hundreds  of  thousands.  Having  now  determined 
the  names  of  the  given  units,  we  read  from  the  left,  and  announce  the  number 
as  two  hundred  sixtv-five  thousand  seventeen. 


FRENCH  SYSTEM  OF  NUMERATION. 

38.  The  separating  of  written  numbers  into  uniform  periods  of  three  figures, 
or  i»laces,  as  cx])lained  above,  is  known  by  its  origin  and  use  as  the  French 
system  of  numeration.     This  is  the  system  invariably  used  in  the  United  States. 

39.  The  Periods  take  their  names  from  the  Latin  numerals,  with  certain 
established  variations,  and  numbers  are  divided  into  orders  of  units  and  into 
periods,  and  are  read  as  shown  by  the  following 

French  Numeration  Table. 


.2  i 


r~      k> 


pacLi  SOh  hoh  p&h 


m 

§ 

S 

a 

.2 

<i-i 

j5 

«M 

^^ 

o 

*^     ■ 

O 

•r^ 

OD 

n 

OD 

S 

cc 

-c 

TZ 

pj 

£ 

<>-■ 

o: 

o; 

<*.! 

Q 

o 

C 

k 

C 

*C 

X 

c 

.S 

a 

D 

0! 

c 

H 

K 

E- 

5 

Mh 

H 

6, 

1 

3 

2. 

7 

4 

13 

a 

ai 

CO 

•v 

S 

o 

O 

■c 

^ 

« 

E- 

Oh 

CO 

•c 

a 

<& 

cc 

J3 

QQ 

c 

H 

s 

«»-i 

o 

o 

01 

H 

^ 

TT 

a 

a 

c 

03 

D 
O 

a 

ZJ 

JS 

t^ 

E- 

0)  O 


6,  9        8       5,  18 


KOTATION    AND    NUMERATION.  5 

The  other  successive  periods  are  called  Quadrillion,  Quintillion,  Sextillion, 
Septillion,  Octillion,  etc.  Commencing  with  the  right  figure,  which  is  called 
units  of  the  first  order,  or  simple  units,  the  orders  of  figures,  or  units,  to  the 
left,  are  called  units  of  the  second  order,  units  of  the  third  order,  fourth,  fifth, 
sixth,  etc. 

ENGLISH  SYSTEM  OF  NUMERATION. 

40.  There  is  in  use  a  system  known  as  the  English  numeration,  which  gives 
to  each  period  after  thousands,  six  places,  or  figures,  instead  of  three  us  given  by 
the  French  numeration.  Numbers  are  divided  into  periods,  and  enumerated  and 
read  by  the  English  numeration  as  shown  by  the  following 


English  Nuiueratiou  Table. 


-% 


T3 
en 

0 
09 

33 

a 
.2 

to 

a 
o 

a 

tn 

3 

■a 

^ 

73 

o 

a 

33 
■X 

a 

00 

a 

o 

a 
a 

3 

<t-l 

o 

<t-i 

'—' 

<t-i 

o 

c 

J 
H 

GQ 

09 

S 

o 

OQ 

in 
T3 

OQ 

09 

-% 

t-i 

i 

a; 

OD 

•a 

«t-i 

1 

13 

a 

u 

o 

bi 

o 

u 

o 

h. 

o 

a 
s 

00 

a 

3 
o 

a 

3 

S 

_o 

•a 

d 

3 

S 

3 
O 
J3 

T3 

3 

3 

OQ 

3 

'S 

fS 

ffi 

H 

H 

w 

H 

s 

ffi 

H 

H 

w 

H 

u 

2, 

5 

7 

3 

1 

8 

1, 

9 

6 

4, 

3 

7 

2, 

Each  period  of  the  higher  orders  has  also  six  places. 

Remark. — The  English  system  of  Numeration  being  of  no  practical  value  to  pupils  in  the 
schools  of  the  United  States,  it  ■will  not  be  hereafter  referred  to. 

41.     The  Arabic  method  of  notation  is  based  upon  the  following 

General  Principles.  —  1.  The  removal  of  any  figure  one  jfloce  toward  the 
left  mnltiplies  its  value  hy  ten ;  two  places,  by  one  hundred ;  three  places,  by  one 
tliousand,  etc. 

2.  The  removal  of  any  figure  one  place  totuard  the  right  divides  its  value  by 
ten  ;  two  places,  by  one  hundred,  etc. 

3.  A  cipher  placed  after  a  significant  figure  multiplies  it  by  ten;  two  ciphers 
so  placed,  muKlplies  it  by  one  hundred,  etc. 


(i 


NOTATION    AND    NLMEUATIOX. 


42.  Write  and  read  ; 

I.  Xine  units  of  the  first  order. 

~.   Five  units  of  the  first  order  and  two  units  of  the  second  order. 

S.  Eight  units  of  the  first  order,  three  of  the  second,  and  one  of  the  first. 

Jf.  Four  units  of  the  fourth  order,  nine  of  the  third,  and  two  of  the  second. 

5.  Two  units  of  the  fifth  order,  nine  of  the  third,  and  seven  of  the  first. 

G.  One  unit  of  the  sixtli  order,  nine  of  the  fourtli,  six  of  the  second,  and  eight 
of  the  first. 

7.  Seven  units  of  the  seventh  order  and  seven  of  the  first. 

S.  Six  units  of  the  eighth  order,  four  of  the  sixth,  seven  of  the  fourtli,  and 
one  of  the  second. 

9.  One  unit  of  the  ninth  order,  two  of  tlie  eighth,  three  of  the  seventh,  four 
of  the  sixth,  five  of  tlie  fifth,  six  of  tlie  fourth,  seven  of  tlie  third,  eight  of  the 
second,  and  nine  of  the  first. 

43.  Express  by  figures  the  following  iiiaubtiv<: 
1.  Sixty-four. 

x'.  One  hundred  forty-eight. 

S.  One  thousand  four  hundred  six. 

^.  Twenty  thousand  twenty-one. 

5.  Three  hundred  sixty-five  thousand. 

6.  Eighty  million  forty-two. 

7.  Ninety  thousand  nine  hundred. 

8.  Fifty  million  fifty-one 

9.  Eighty-seven  billion  seven  thousand  twelve. 

i".  Ninety-seven  million  ninety-seven  thousand  ninety-seven. 

II.  Twenty-one  million  twenty-five. 

1^.  Sixteen  billion  sixteen  million  sixteen. 
13.  Six  hundred  eighty-nine  thousand  nine  hundred  seven. 
H.  Nineteen  billion  five  hundred  forty-one  million  eh-ven  thousand  eleven. 
I'j.  Twenty-seven  quiutillion  eighty-one  quadrillion  two  trillion  seven  hundred 
sixty  billion  one  million  two. 

44.  Point  off  into  periods,  numerate,  and  read  the  following  numbers: 


1. 

380. 

V. 

1341. 

o. 

1240G. 

4. 

79001. 

5. 

872403. 

fi. 

001008. 

i   . 

4G81005. 

45.  Writ 

1. 

920. 

> 

1146. 

.?. 

3070. 

S. 

77010016. 

9. 

200020. 

10. 

140024G780. 

11. 

2100211. 

12. 

5800092. 

13. 

34307001. 

U- 

lOOOlOOOlOUd. 

6103G. 


iu  wiirils  mill  read  the  following  numlx-r 
I   n.     50415.  I 

6'.     100000. 

7.  521469. 

8.  201012. 


iJ. 

987000460000. 

iij. 

27510304050. 

17. 

11002200330044. 

18. 

2234507890. 

19. 

40122555003. 

20. 

621438001240709. 

21. 

12345325500001503. 

9. 

1406250. 

10. 

54790207. 

11. 

1021714. 

12. 

5790r)7359. 

(^ 


a 


NOTATION'    AND    NUMERATION.  7 

ROMAN   NOTATION  AND   NUMERATION. 

46.  Bj  combining,  according  to  certain  princiiiles,  the  letters  used  in  this 
method  of  writing  numbers,  any  number  can  be  expressed. 

Principles. — 1.   Repeating  a  letter  repeats  its  value. 
Thus,  I  =  one,  II  =  two,  X  =  ten,  XX  =  twenty. 

2.  If  a  letter  of  any  value  is  annexed  to  one  of  greater  value,  the  sum  of  tlie  two 
values  is  indicated  ;  if  a  letter  of  any  value  is  prefixed  to  one  of  greater  value,  the 
difference  of  their  values  is  indicated. 

Thus,  XI  denotes  X  +  I  =  eleven,  IX  denotes  X  —  I  =  nine. 

3.  A  dash  — placed  over  a  letter  multiplies  its  value  by  one  thousand. 

Thus,  V=  five,  V=  five  thousand,  CD  =  four  hundred,  CD  =  four  hundred 
thousand,  LXVII  =  sixty-seven,  LXVII  =  sixty-seven  thousand. 


Table  of  Roman  Numerals  with  Arabic  Equivalents. 


I,- 

11, 

III, 

IV, 

V, 

VI, 

VII, 

VIII, 

IX. 

X, 

XI, 


1. 

2. 
3. 
4. 
5. 

6. 

7. 

8. 

9. 
10. 
11. 


XII, 

XIII, 

XIY, 

XV, 

XVI, 

XVII, 

XVIII, 

XX,  /  ^ 
XXX, 

XL, 


12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
30. 
40. 


L, 

LX, 

LXX, 

LXXX, 

XC, 

c, 

CC, 

CCC, 

CD, 

D, 

DC, 


50. 

60. 

70. 

80. 

90.  ■ 
100. 
200. 
300. 
400. 
500. 
600. 


DCC, 

700. 

DCCC, 

800. 

CM, 

900. 

M, 

1000. 

MM, 

2000. 

X, 

10000 

L, 

50000. 

c. 

100000 

D, 

500C00 

M, 

1000000 

47. 

XCII. 
XXVII. 
XXIX. 
CLX. 

48. 
42. 
111. 
66G, 
1125. 
7000. 
,11451. 
997. 
56104. 
3001. 


Eead  the  following  expressions: 

CCXVII.  CMXIX. 


DCV. 

DCCX. 

CMXXV. 


MCLXXIX. 

MCDXCII. 

MDCCLVI. 


DLXX. 


DCCXLV. 

MDq.  . 

^DCCCLXXXVIII. 


Express  by  the  Roman  system  the  following  numbers 


7454. 
8709. 
62550. 
1620. 
399. 
"^-^5406." 
48250. 
3700. 
2865. 


1629. 
^1889. 
"^60012. 

3658. 

175400. 

1761. 

1887. 

1000000. 

20000. 


45450. 

19015. 

1111. 

6057. 

3113. 

90055. 

805000. 

365. 

1515. 


6059. 
-*21021. 
4888. 
9()909. 
5168. 
1890. 
1775. 
1893. 
1900. 


ADDITION. 


ADDITION. 


49.  Addition  is  the  process  of  combining  several  numbers  into  one  equiva- 
lent number. 

50.  The  Sum  or  Amount  is  tlie  result  obtained  by  the  addition  of  two  or 
more  numbers. 

51.  The  Sign  of  Addition  is  +,  and  is  called  Plus,  which  signifies  more. 
When  placed  between  two  numbers  or  combinations  of  numbers,  it  indicates 
their  addition;  as,  5  +  2  is  read  5  plus  2,  and  shows  that  5  and  2  are  to  be  added. 

52.  The  Sign  of  Equality  is  =  .  When  placed  between  two  numbers  or 
combinations  of  numbers,  it  indicates  that  there  is  no  difference  in  their  value; 
thus,  5  4-2  =  T,  is  read  5  plus  2  equals  7..  and  indicates  that  the  value  of  7  equals 
the  value  of  the  sum  of  the  numbers  at  the  left  of  the  sign  of  equality. 

53.  Carrying  the  Tens  is  the  process  of  reserving  the  tens  and  adding 
them  with  the  next  column. 

54.  Principles. — 1.  Only  like  numbers  and  like  unit  orders  can  be  added 
one  to  another. 

2.  The  sum  or  amount  contains  as  many  units  as  all  the  numbers  added. 

3.  The  sum  or  amount  is  the  same  in  whatever  order  the  numbers  be  added. 

55.  Addition  is  the  Reverse  of  Subtraction  and  may  be  proved  by  it; 
as,  5  +  2  =  7.  Xow  if  7  be  diminished  by  5,  the  result  Avill  be  2,  while  if  7  be 
diminished  by  2,  the  result  will  be  5. 

56.  Numbers  are  written  for  addition  either  in  vertical  or  horizotital  order. 

57.  General  Kules. — l.  If  the  sum  of  two  numhers  and  one  of  the 
numbers  be  given,  the  unknown  number  may  be  found  by  taking  the 
given  nuTtiber  from  the  sum. 

2.  If  the  suTti  of  several  nunibers  and  all  of  the  numbers  but  one  be 
given,  the  unknown  number  may  be  found  by  subtracting  the  sum  of 
those  given  from  the  sum  of  all  the  numbers. 

Notes  to  Teacher.  —  1.  Classes  should  have  frequent  and  extended  drill  in  rapid  mental 
addition. 

2.  The  following  table  is  given  simply  to  facilitate  class  drill,  preparatory  to  work  in  rapid, 
addition. 


ADDITION. 


6  +   6  = 

ij  +    7  = 

6  +  8  = 

6  +  9  = 

6  +  10  = 

6  +  11  = 

6  +  12  = 

6  +  13  = 

6  +  14  = 

6  +  15  = 

6  +  16  = 

6  +  17  = 

6  +  18  = 

G  +  19  = 

6  +  20  = 

6  +  21  = 

6  +  22  = 

6  +  23  = 

6  +  24  = 

6  +  25  = 

7  +  7  = 


7+9 
7  +  10 
7  +  11 
V  +  12 
7  +  13 
7  +  14 
7  +  15 
7  +  16 
7  +  17 
7  +  18 
7  +  19 
7  +  20 
7  +  21 
7  +  22 
+  23 
+  24 


7 
7 

7  +  25  = 

8  +  8  = 
8  +  9  = 
8  +  10  = 
8  +  11  = 


12 
13 
14 

15 
16 

ir 

18 
10 
20 
21 
22 
23 
24 
25 
20 
27 
28 
29 
30 
31 

14 
15 
16 


=  r 


18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 

16 

ir 

18 
19 


8  + 

8  + 

8  + 

8  + 

8  + 

8  + 

8  + 

8  + 

8  + 

8  + 

8  + 

8  + 

8  + 

8  + 


Addition 

12  =  20 

13  =  21 

14  =  22 

15  =  23 

16  =  24 

17  =  25 

18  =  2G 

19  =  27 

20  =  28 

21  =  29 

22  =  30 

23  =  31 

24  —  32 

25  =  33 


9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

9 

+ 

10 

+ 

10 

+ 

10  + 

10  + 

10 

+ 

10 

+ 

10 

+ 

10  + 

10  + 

10  + 

10  + 

10 

+ 

9  =  18 

10  =  19 

11  =  20 

12  =  21 

13  =  22 

14  =  23 

15  =  24 

16  =  25 

17  =  20 

18  =  27 

19  =  28 

20  =  29 

21  =  30 

22  =  31 

23  =  32 

24  =  33 

25  =  34 


10  =  20 

11  =  21 

12  =  22 

13  =  23 

14  =  24 

15  =  25 

16  =  26 

17  =  27 

18  =  28 

19  =  29 

20  =  30 

21  =  31 


Table  for  Class  Drill. 

10  +  22  =  32  13  +  22  =  35 
10  +  23  =  33  13  +  23  =  36 
10  +  24  =  34  13  +  24  =  37 
10  +  25  =  35  13  +  25  =  38 


11  +  11  = 
11  +  12  = 
11  +  13  = 
11  +  14  = 
11  +  15  = 
11  +  16  = 
11  +  17  = 
11  +  18  = 
11  +  19  = 
11  +  20  = 
11  +21  = 
11  +22  = 
11  +  23  = 
11  +  24  = 
11  +  25  = 


22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 


12  +  32 

12  +  13 
12  +  14 
12  +  15 
12  +  16 
12  +  17 
12  +  18 
12  +  19 
12  +  20 
12  +  21 
12  +  22 
12  +  23 
12  +  24 
12  +  25 


24 
25 
26 
27 
28 
29 
.30 
31 
32 
33 
34 
35 
36 
37 


13  +  13 
13  +  14 
13  +  15 
13  +  16 
13  +  17 
13  +  18 
13  +  19 
13  +  20 
13  +  21 


26 
27 
28 
29 
30 
31 
32 
33 
34 


14  +  14 
14  +  15 
14  +  16 
14  +  17 
14  +  18 
14  +  19 
14  +  20 
14  +  21 
14  +  22 
14  +  23 
14  +  24 
14  +  25 


28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


15  +  15 
15  +  16 
15  +  17 
15  +  18 
15  +  19 
15  +  20 
15  +  21 
15  +  22 
15  +  23 
15  +  24 
15  +  25 


30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
4C 


16  +  16 
16  +  17 
16  +  18 
16  +  19 
16  +  20 
16  +  21 
16  +  22 
16  +  23 
16  +  24 
16  +  25 


32 
33 
34 
35 
36 
37 
38 
39 
40 
41 


17  +  17 
17  +  18 
17  +  19 
17  +  20 
17  +  21 


34 
35 
36 
37 
38 


17  -^  22  =  3^ 
17  +  23  =  40 
17  +  24  =  41 
17  +  25  =  42 


18 
18 
18 
18 
18 
18 
IS 
18 


+  18  =  36 
+  19  =  37 
+  20  =  3& 
+  21  =  39 
+  22  =  40 
+  23  =  41 
+  24  =  42 
+  25  =  43 


19 
19 
19 
19 
19 
19 
19 


+  19  =  38 
+  20  =  39 
+  21  =  40 
+  22  =  41 
+  23  =  42 
+  24  =  43 
+  25  =  44 


20 
20 
20 
20 
20 
20 


+  20  =  40 
+  21  =  41 
+  22  =  42 
+  23  =  43 
+  24  =  44 
+  25  =  45 


21 
21 
21 
21 
21 


+  21  =  42 
+  22  =  43 
+  23  =  44 
+  24  =  45 
+  25  =  46 


22 
22 
22 
22 


+  22  =  44 
+  23  =  45 

+  24  =  46 
+  25  =  47 


+  23  =  4e 
+  24  =  47 

+  25  =  4& 


24 
24 


+  24 
+  25 


48 
49 


+  25  =  50 


10  ADDITION. 

MENTAL   KXERCISES. 


58. 

1 

Add 
{2.) 
5 

{3.) 

1 

a) 

0 

(5.) 

0 

11 

Cr.) 

V2 

25 

2 

6 

3 

4 

•  5 

14 

14 

50 

3 

^ 

5 

6 

4 

13 

18 

15 

-t 

8 

•  1^ 
1 

8 

8' 

15 

15 

25 

5 

9 

9 

10 

6 

12 

10 

10 

6 

10 

11 

1-2 

12 

18 

20 

25 

1 

11 

13 

1-t 

10 

i: 

14 

15 

8 

12 

15 

IG 

14 

20 

1€ 

10 

59.  There  are  three  methods  of  addition  in  common  use,  viz. :  tlie  Elementary 
vipfhofl.  the  Besidt  method,  and  the  Group  method. 

Remarks. — 1.  These  methods  of  additibn  are  recommended  to  be  taught  in  their  order  to 
pupils  in  elementary  -work;  the  first,  as  soon  as  mastered,  should  be  abandoned  for  the  second, 
and  the  second  in  its  turn,  when  mastered,  abandoned  for  the  third. 

2.  Daily  drill  in  the  third  method  is  urgently  advised  with  all  pupils  during  the  entire 
period  of  their  study  of  Arithmetic.  Too  much  importance  can  scarcely  be  attached  to  this 
suggestion. 

60.  The  Elementary  Method  of  Addition. 

Example.— Add  32,  71,  25,  48,  90,  12,  and  03. 

OPERATION.        ExPLANATiOK.— Having  arranged  the  numbers  so  that  units  of  like  orders 
„.^  stand  directly  under  each  other,  begin  with  the  last  figure  in  the  right-hand, 

or  units'  column,  and  add  upward  as  follows  :  3  and  2  are  5,  5  and  8  are  13, 
'  ^  13  and  5  are  18,  18  and  1  are  19,  19  and  2  are  21.     Having  thus  obtained  the  sura, 

25  place  the  1  beneath  the  line,  in  units'  column,  and  treat  the  2  as  a  part  of  the 

48  second,  or  tens'  column,  which  add  upVard  as  before  ;  thus,  2  and  6  are  8, 

90  8  and  1  are  9,  9  and  9  are  18,  18  and  4  are  22,  22  and  2  are  24,  24  and  7  are  31. 

19  31  and  3  are  34.     Having  obtained  the  sum,  write  it  in  full  at  the  left  of  the 

^_  figure  1  before  written,  and  the  result  is  341,  the  numerical  expression  of  the 

'_  sum  of  the  numbers  added. 

Q^j  To  Prove. — Add  the  columns  downward  ;  if  the  two  results  agree,  the 

work  is  presumed  to  be  correct. 


61.     The  Eesult  Method  of  Addition. 
Example.— Add  32,  71,  25,  48,  90,  12,  and  03. 


OPERATIOK. 
32 

.  71 
25 
48 
90 
12 
53  To  Prove.— .\dd  the  columns  downward. 

341 


ExPLAKATioN. — Beginning  as  before,  with  the  lower  figure  in  units'  column, 
name  the  result  only  of  each  successive  addition,  thus:  3,  5, 13, 18, 19,  21  ;  then, 
as  before,  write  the  1  beneath  the  line  in  units'  column  and  carrying  the  2  to 
tens'  column  as  a  part  of  it,  add  upward,  thus  :  2,  8,  9,  18,  22.  24,  31,  34 ;  as 
before,  write  34  at  the  left,  and  the  result  is  341,  the  same  as  before. 


ADDITION.  11 

6'2.     The  Group  Method  of  Addition. 

Example.— i.  Add  32,  71,  25,  48,  90,  12,  and  63. 

■OPERATION.  Explanation.  —  Treat  the  same  numbers  thus  :   add  upward  ;  3,  13,  21  ; 

32  grouping  2  and  8  for  10  to  add  to  8,  making  13,  and  5,  1,  and  2  for  8,  to  add  to 

f.-.  I  o  13,  making  21.     Having  written  the  1  beneath  the  line,  in  units'  place,  carry  the 

_  C  2  or  2  tens,  to  its  column,  and  again  add  ;  2,  8,  18,  24,  34  ;  grouping  2  and  6 

^^  ^  for  8,  9  and  1  for  10,  4  and  2  for  6,  and  7  and  3  for  10  ;  then  write  the  result 

48  ^  in  full  as  before. 

90  V  10  To  Provk.  —  Review  the  first  column  by  adding  downward  ;  8,  18,  21  ; 

12  )  grouping  2,  1,  and  5  for  8,  8  and  2  for  10,  to  add  for  18,  and  to  this  add  the 

63  remaining  figure  3,  for  21,  the  same  result  as  before.     Tlien  review  the  second 

column  by  adding  downward  ;  10,  16,  26,  34  ;  grouping  3  and  7  for  10,  2  and  4 

341  for  6,  9  and  1  for  10,  and  6  and  2,  for  8,  with  the  same  result. 

•Example.— ^   Add  3417,  2140,  439,  7164,  1538,  5046,  6116,  8735,  971,  4880, 

1263,  9270,  192,  and  634. 

OPERATION.  Explanation. — Beginning  with  the  lower  unit  figure  add  upward;  10, 15,  35, 

341 '^  1  00,  grouping  4,  2,  3,  and  1  for  10,  which  added  to  5  gives  15  ;  grouping  6,  6,  and 

2140  ^  ^*^"'  ^^  ^"  ^^^^  ^°  ^^  obtaining  35  ;  and  grouping  4,  9,  and  7  for  20  to  add  to  35 

439  I  ^"^  ^5  ^^^^  result.     Write  the  units'  figure  5  in  its  place,  and  carrying  the  tens' 

7164  I  figure  5  to  its  column  proceed  thus  :  8,  24,  38,  48,  56,  60,  70,  grouping  the  5 

1538  1  carried  and  3  for  8  ;  9  and  7  for  16  to  add  to  8  for  24  ;  6  and  8  for  14  to  add  to 

5046  I  20  24  to  make  38  ;  7  and  3  for  10,  making  48  ;  1,  4,  and  3  for  8,  making  56  ;  6,  3, 

6116  J  iind  1  for  10,  making  06,  to  which  we  add  the  4  for  70,  the  result.     Write  the 

8735  cipher  of  the  70  at  the  left  of  the  unit  figure  already  written  beneath  the  line 

971 1  and  carrying  the  7  to  the  third,  or  hundreds'  column  group  as  before;  16,  26,  36, 

4880  48,  58,  grouping  upward  thus  :  7,  6,  1,  2  =  16  ;  2,  8  =  10  ;  9,  1  ==  10  ;  7,  5  =  12  ; 

1263  I  1^  1,  4,  1,  4  =  10.     Write  the  8  in  hundreds'  column,  and  carrying  the  5  to  thou- 

9270  I  sands'  column,  group  15,  27,  39,  49,  51.     5,  9,  1  =  15  ;  4,  8  =  12  ;  6,  5,  1  =  12  ; 

192  I  7.  3  =  10,  and  adding  2  write  the  result,  51,  at  the  left  of  the  figures  before 

634  J  written,  thus  obtaining  51805  the  numerical  expression  of  the  sum  of  the  num- 

51805  ^^''^^  added.                         • 

Prove  by  adding  downward,  grouping  as  illustrated  above. 

Remark. — Practice  in  grouping  will  lead  to  great  proficiencj',  and  after  the  pupil  becomes 
.somewhat  skilled,  he  should  be  encouraged  to  skip  about  somewhat  along  the  column,  in 
order  to  select  those  numbers  which  can  be  most  conveniently  grouped.  Ordinarily  thorough 
<lrill  in  the  addition  table  will  greatly  assist  in  grouping,  and  multiples  of  the  nine  digits  can 
be  added  with  ease.  Excci^t  with  very  bright  pupils,  groups  greater  than  25  are  not  to  be 
recommended. 


HORIZONTAL  ADDITION. 

63.  Xumbers  "when  "written  in  liorizontul  order,  as  in  invoices  and  otiier 
business  forms,  may  be  added  Avithont  being  re-written  in  vertical  columns. 

Remarks. — 1.  In  adding  numbers  written  horizontally,  more  care  is  requisite  that  the  units 
added  shall  be  of  like  order,  and  greater  certainty  of  correctness  can  be  had  ])y  adding  first 
from  left  to  right,  and  then  from  right  to  left. 

2.  The  group  method  may  be  employed  vv^ith  equal  advantage  where  numbers  are  written 
liorizontally. 


12  ADDITION. 

MKXTAI^   KXKKCISES. 

(J4-.     Add  from  loft  to  right,  and  review  from  right  to  left. 


1.  5.  3,  6,  1,  8,  2,  7,  9,  4. 

2.  21,  56,  12,  93,  47,  60,  17. 
S.  (S(j,  29,  5,  14,  71,  19,  2,  11. 
4.  149,  865,  73,  40,  5,  13,  502. 
o.  365.  10,  88,  46,  200,  175,  95. 


15,  23,  36,  18,  25,  53,  92. 
7.     11,  85,  315,  125,  111,  206. 
S.     8,  42,  87,  20,  112,  108,  94,  12s. 
9.     6L,  400,  1,  126,  25,  440. 
10.     25,  50,  511,  3,  209,  8.  804. 


WKITTKN   KXKKCISES. 

65.  Copy,  and  add  from  left  to  right;  review  from  riglit  to  left,  preserving; 
results. 

1.  510,  297,  69,  841,  638,  203,  40,  7,  700,  28,  9. 

J.  1260,  2700,  408,  9206,  51,  7240,  27,  1620. 

S.  8809,  1492,  1000,  20,  1,  504,  6620,  7506,  10. 

Jf.  50000,  20000,  8900,  21050,  47800,  14090. 

5.  76030,  20500,  38037,  69000,  81,  107,  2,  19975. 

6.  346211,  218040,  173508,  973200,  701001,  555555. 

7.  604000,  181523,  51,  19406,  200,  309,  5,  2,  8000. 

5.  2463911,  7054133,  4444044,  1371005,  6090400. 
9.     8500500,  1035660,  5000000,  2987400,  7020319. 

10.     416,  49,  2,  7967400,  81,  307,  21021,  190200,  40,  3. 

Remark. — Horizontal  addition  is  rarely  practiced  with  numbers  containing  more  than  four 
or  five  figures.  It  may  sometimes  be  employed  to  advantage  in  adding  dollars  and  cents;  in 
such  cases  it  is  best  to  omit  the  dollar  sign;  as,  for  $5.25  write  5.25. 

66.  Copy  and  add  horizontally;  review  and  preserve  results. 
1.     5.25,  8.17,  11.40,   1.82,  16.02,  90.70. 

^.  146.24,  9.11,  210.10,  46.98,  5.50,  108.12,  4.75. 

3.  26.53,  92,   5.71,  108.97,  29.33,   150,  46.07,  19,   76. 

Jf.  231.45,  50,  75,  19.78,  40,  50,  63,  TOO. 

o.  63,  51,  87,  25,  75,  18,  .09,  95,  1.25,  6. 

6.  278.19,   105.29,  80.50,  19.93,  52,  1. 

7.  29.30,  403,  51,  73,  1.14,  90,  300,   1.25. 

8.  1.13,  9.25,  14,  27.16,  5.01,  8,  25,  1.75. 

9.  87.50,  125,  36.21,  9.90,  14.75,  16,  25.25. 

10.  117.82,  7.71,   19.03,  15,  49.55,  87.08. 

11.  5.40,  88,  35,  90,  112.50,  45.95,  111.50. 

12.  100,  79.22,  50.08,  2.25,  7.75,  10,  3,  8.24. 

13.  216.24,  92,  15,  .06,  138.50,  2.38,  9.25. 

Rem.\rk. — The  teacher  may  give  other  examples  of  the  same  kind  ;  he  will  find  extensive 
drill  in  such  work  of  great  value  to  all  grades  of  pupils,  in  developing  accuracy  and  rapidity. 

EXAMPLES  FOR   PKACTICE. 

67.  1.  A  grocer's  sales  were,  for  Monday,  8241;  Tuesday,  $306;  Wednesday, 
8523  ;  Thursday,  $438  ;  Friday,  $497  ;  on  Saturday  his  sales  amounted  to  $27 
more  than  the  sales  of  the  first  three  days  of  the  week.  What  were  his  total 
sales  during  the  week? 


ADDITION.  13 

2.  A  planter  shipped  eleven  bales  of  cotton,  weighing  respectively  492,  504 
523,  487,  490,  500,  516,  499,  512,  511,  and  496  pounds.     What  was  the  aggregate 
weight  of  the  shipment  ? 

3.  A  jiortable  saw-mill  cut  lumber  for  the  six  working  days  of  a  week,  as 
follows  :  On  Monday,  5116  feet ;  Tuesday,  4900  feet ;  Wednesday,  5750  feet ; 
Thursday,  6100  feet ;  Friday,  4580  feet ;  and  on  Saturday,  6754  feet.  AVhai 
amount  of  lumber  did  the  mill  cut  during  the  week? 

Jf..  Find  the  sum  of  four  units  of  the  second  order  and  five  of  the  first;  eight 
of  the  fifth,  three  of  the  third,  and  nine  of  the  second;  seven  of  the  sixth,  one 
of  the  fifth,  and  two  of  the  third;  one  of  the  eighth,  nine  of  the  tliird,  seven  of 
the  second,  and  six  of  the  first;  four  of  the  fifth,  three  of  the  fourth,  and  nine 
of  the  third;  Aa'c  of  the  tenth,  one  of  the  ninth,  four  of  the  seventh,  eight  of 
the  third,  two  of  the  second,  and  one  of  tlie  first, 

5.  Find  the  sum  of  sixty-nine  thousand  five  hundred  seven,  one  thousand  six 
hundred  twenty-two,  one  hundred  fifty-six  thousand  seventy-six,  ninety-nine 
thousand  nineteen,  forty-one  million  eighty-seven  thousand  five,  three  hundred 
twenty-five  million  sixteen  thousand  eight  hundred  eighty-eight,  six  billion 
ninety-one  million  four  thousand  two  hundred  fifty-six. 

6.  The  British  House  of  Lords  was,  in  1884,  comprised  of  4  princes,  23  dukes, 
19  marquises,  139  earls,  32  viscounts,  26  bishops,  and  272  barons.  How  many 
members  in  all  ? 

7.  In  1883,  there  arrived  and  settled  in  the  United  States,  immigrants:  Germans, 
192,000;  English,  100,200;  Canadians,  65,100;  Irish,  64,400;  Scandinavians, 
52,200;  Italians,  32,500;  miscellaneous,  92,700.  What  was  the  total  number  of 
immigrants  ? 

S.  The  British  national  debt  in  March,  1883,  was  :  Consols,  £699,053,100  ; 
Bank  debts,  £13,645,900;  Annuities,  £27,570,900;  Exchequer  Bills,  £8,754,400; 
Treasury  Bills,  £5,431,000  ;  Savings  Banks,  £1,804,400  ;  and  the  local  debt, 
£163,501,000.     What  was  the  total  debt  in  pounds  sterling  ? 

9.  In  1866,  the  V.  S.  collected  as  revenue  from  Customs,  $179,046,651.58  ; 
from  Internal  Revenue,  $309,226,813.42;  from  Direct  Taxes,  $1,974,754.12; 
from  the  Public  Lands,  $665,031.03  ;  and  from  other  sources,  $29,036,314.23. 
What  was  the  total  government  revenue  collected  that  year? 

10.  The  British  government  collected  as  revenue  in  1882  :  From  Customs, 
£19,300,000;  from  Excise,  £27,230,000;  from  Stamps,  £11,145,000;  from  Land 
Tax,  £2,775,000;  from  Income  Tax,  £11,662,000;  from  Post  Office,  £7,150,000, 
from  Telegraphs,  £1,650,000;  from  Crown  Lands,  £380,000;  from  Interest, 
£1,180,000  ;  from  miscellaneous  sources,  £4,725,000.  What  was  the  total  reve- 
nue of  the  British  government  for  that  year? 

11.  The  dwarf,  Borowlaski,  was  only  39  inches  in  height ;  Tom  Thumb,  31  ; 
Mrs.  Tom  Thumb,  32  ;  Che-Mah,  of  China,  25  ;  Lucia  Zarate,  of  Mexico,  20  ; 
and  Gen.  Mite,  21.     What  was  the  combined  height  of  the  six? 

12.  The  firm  of  Davis  &  Drake  own  land  valued  at  $39,750;  lumber,  $68,125; 
notes,  $21,700  ;  book  accounts,  $17,291  ;  machinery,  $13,250  ;  cash  in  bank, 
$14,238  ;  cash  on  hand,  $4,332.     What  is  the  i)roperty  value  of  the  firm  ? 


14 


ADDITION. 


13.  In  1880,  tliere  were  women  workers  in  the  United  States  as  follows:  artists, 
2,061;  autliors,  320  ;  barbers,  2,902  ;  dressmakers,  281,928  ;  journalists,  288  ; 
lawvers,  75;  musicians,  13,181;  phj'sicians,  2,432;  preachers,  105  ;  printers, 
3,456;  tailors,  52,098;  teachers,  154,375.     How  many  women  workers  in  all  ? 

Remark. — The  three  following  problems  can  be  properly  iised  by  the  teacher  for  drill  in 
group  adding. 

IJf..  The  population  of  the  United  States,  by  the  census  of  1880,  is  as  follows: 


Ala., 1,262,505 

Alaska, 30,000 

Ariz., 40,440 

Ark., 802,525 

Cal., 864,694 

Colo.,-'. 194,327 

Conn.. 622,700 

Dak., 135,177 

Del., 146,608 

D.  C, 177,624 

Fla., 269,493 

Ga., 1,542,180 

Idaho, 32,610 

111., 3,077,871 

Ind., 1,978,301 

Ind.  T 70,000 


Kans.,.-. 996,096 

Ky., 1,684,690 

La., 939,946 

Me., 648,936 

Md., 934,943 

Mass., 1,783,085 

Mich.,. 1,636,937 

Minn., 780,773 

Miss., 1,131,597 

Mo., 2,168,380 

^[ont.  T 39,159 

Xebr., 452,402 

Xev., 62,206 

X.  H., 346,991 

X.  J., 1,131,116 

X.  Mex.,- 119,505 


X.  Y.,. 5,082,871 

X.  C, 1,399,750 

Ohio,... .3,198,062 

Oregon,- 174,768 

Pa., 4,282,891 

R.  L, 276,531 

S.  C, *--    995,577 

Teun.,. ...1,542,359 

Tex., 1,591,749 

Utah, 143,963 

Vt., 332,286 

Ya., 1,512,565 

Wash.  T., 75,116 

W.  Yu., 618,457 

Wis., '.1,315,497 


Wvo. 


20,789 


Iowa, .1,624,615 

What  was  the  total  population  ? 

15.  Tiie  area  uf  the  United  States,  in  square  miles,  is  as  follows  : 


Ala., 51,540 

Alaska, 531,409 

Ariz., 112,920 

Ark., 53,045 

Cal., 155,980 

Colo., 103,645 

Conn., 4,845 

Dak.,     .147,700 

Del., 1,960 

D.  C, .-  60 

Fla., 54,240 

Ga., 58,980 

Idaho, 84,290 

111., -.--   56,000 

Ind., .--  35,910 

Ind.  T., 69,830 

Iowa, 55,475 

What  is  the  total  area  ? 


Kans., 81,700 

Ky., .  40,000 

La., 45,420 

Me., 29,895 

Md.,.. 9,860 

Mass., 8,040 

Mich., 57,430 

Minn., 79,205 

Miss., 40,340 

Mo., 68,735 

Mont.  T.,/ 145,310 

Xebr., 76,185 

Xev., 109,740 

X.n., 9,005 

X.  J., 7,455 

X.  Mex., 122,460 


X.  Y.,... 47,620 

X.  C, 48,580 

Ohio, 40,760 

Oregon, 94,560 

Pa., 44,985 

R.  L, 1,085 

S.  C, 30,170 

Tenn., 41,750 

Tex., 262,290 

Utah, 82,190 

Yt.,.-.. 9,135 

Ya., 40,125 

Wash.T., 66,S80 

W.  Ya., --.   24,645 

Wis., 54,450 

Wvo.  T., 97,575 


ADDITION. 


15 


16.    For  state  tax  of  1888 
were  assessed  as  follows  : 

Albany, 186,606,307 

Alleghany, 14,395,123 

Broome, 21,383,568 

Cattaraugus, 16, 050, 985 

Cayuga, 30,631,548 

Chautauqua, 25,649,740 

Chemnng, 18,718,275 

Chenango, 1 7,982,340 

Clmton, 9,766,255 

Columbia,  . . . . 20,984,129 

Cortland, 11,108,469 

Delaware, . 13,921,534 

Dutchess, 44,532,280 

Erie, 127,763,104 

Essex, 10,515,260 

Fulton, 8,383,735 

Franklin, 8,026,235 

Genesee, 21,384,810 

Greene, 13,760,299 

Hamilton, 1,157,600 

Herkimer, 23, 739,092 

Jefferson, 23,638,204 

Kings, 342,116,976 

Lewis, 9,039,285 

Livingston, 25,395,180 

Madison, 19,797,535 

Monroe, 85,964,190 

Montgomery, 23,877,638 

New  York, 1,500,550,825 

Niagara, 26,097,826 


the  several  Counties  of  the  State  of  New  York 


Oneida, .  .%58,146,279 

Onondaga, 63,265,536 

Ontario, 29,389,870 

Orange, 42,953,974 

Orleans, 14,816,445 

Oswego,  .... 23,655,679 

Otsego, 22,544,650 

Putnam, 7,483,530 

Queens, 44,464,675 

Rensselaer, 60,545,955 

Rockland, 13,394,485 

Richmond, f2,271,105 

Saratoga, 23,189,435 

Schenectady, : 12,772,451 

Schoharie, 10,297,219 

Schuyler, 7,248,620 

Seneca,  _ .  • 15,347,372 

St.  Lawrence, 24,476,078 

Steuben, 22,776,074 

Suffolk, 17,262,646 

Sullivan, 5,427,300 

Tioga, 12,084,525 

Tompkins, 15,450,670 

Ulster, 25,443,000 

Warren, 6,555,175 

Washington, 22,501,173 

Wayne, . 25,404,569 

Westchester, 82,375,217 

Wyoming, 14,922,986 

Yates, 12,721,716 


What  was  the  total  assessed  value  of  the  State  that  year  ? 


IC  SUBTRACTION. 


SUBTRACTION. 

68.  Subtraction  is  the  process  of  finding  the  difference  between  two 
numbers. 

69.  Tlie  Subtrahend  is  the  number  to  be  subtracted. 

70.  Tl»e  Minuend  is  tlie  number  from  which  the  Subtrahend  is  to  be 
subtracted. 

71.  Tlif  Difference  or  Remainder  is  the  result  obtained  by  subtracting 
one  number  from  another. 

72.  TIk-  Siffu  of  Subtraction  is  — .     It  is  called  Minus  and  signifies  less. 

When  the  sign  of  subtraction  i^  placed  between  two  numbers  it  indicates  that  the  number 
placed  after  it  is  to  be  taken  from  the  one  before  it. 

73.  The  Complement  of  a  Number  is  the  difference  between  it  and  a 
unit  of  the  next  liigher  order. 

Thus  the  complement  of  7  is  3,  because  1  ten,  the  unit  of  the  next  higher  order,  diminished 
by  7  =  3.  Again,  the  complement  of  36  is  64,  because  the  unit  of  the  next  higher  order, 
1  hundred,  or  100,  diminished  by  36  =  64. 

74.  Principles. — 1.  Only  like  numbers  and  units  of  the  same  order  can  be 
subtracted,  one  froui  the  other. 

~.    The  sum  of  the  subtrahend  and  the  remainder  must  he  equal  to  the  minuejid. 

75.  General  Relation  of  Terms  in  Subtraction. 

1.  The  Minuend  —  the  Subtrahend  =  the  Remainder. 

II.  The  Minuend  —  the  Remainder  =  the  Subtrahend. 

III.  The  Subtrahend  -\-  the  Remainder  =  the  Minuend. 

76.  General  Rules. — 1.  If  the  minuend  and  subtrahend  be  given, 
the  remainder  may  be  found  by  subtracting  the  subtrahend  from  the 
minuend. 

2.  If  the  minuend  and  remainder  be  given,  the  subtrahend  may  be 
found  by  sid)tracting  the  remainder  from  the  Jivinuend. 

3.  If  the  remainder  and  subtrahend  be  given,  the  rtiinuend  may  be 

found  by  adding  the  remainder  to  the  subtrahend. 

* 

77.  To  ProYe  Subtraction. — Add  the  remainder  to  the  subtrahend  :  if  tlie 
3um  equals  the  tnintiend,  the  loork  is  correct. 


SUBTRACTION. 


17 


Subtraction  Table. 

78.     Find  the  difference,  mentally,  between 


13 

and 

6 

16   and   13 

19   an 

d   10 

21 

and 

16 

24 

and 

5 

12 

i( 

7 

16    "        13 

19 
19 

11 
'   12 

31 
21 

17 
18 

24 
24 

is 

6 

12 

' 

8 

* 

7 

12 

i  i 

9 

17   an 

17 

17 

d   3 

4 
5 

19 
19 
19 

13 
14 
15 

22 

and 

3 

24 

ss 

8 

13 

and 

4 

23 
22 

4 
5 

24 
24 

i  i 

i  ( 

9 

10 

13 

a 

5 

17 

6 

19 

16 

22 

(( 

6 

24 

i  i 

11 

13 

is 

6 

17 

I               rv 
< 

22 
22 

(( 

7 

8 

9 

10 

11 

24 

ii 

12 

13 

i( 

7 

17 

8 

20   and   3 

(( 

24 

ss 

13 

13 

a 

8 

17 

9 

20 

4 

22 

(C 

24 

ss 

14 

13 

a 

9 

17 
17 

10 
11 

20 
20 

'    5 

6 

22 
22 

(C 

ti 

24 
24 

ss 

ss 

15 

14 

and 

5 

16 

14 

a 

6 

17 

13 

20 

'    7 

22 

IS 

12 

24 

ss 

17 

14 

a 

7 

17 

13 

20    ' 

8 

22 

ss 

13 

24 

it 

18 

14 

a 

8 

17 

14 

20 
20 
20 
20 

9 

10 

11 

'   12 

22 

ss 

14 

24 

i  i 

19 

14 

14 

9 
10 

18   ar 
18    ' 

id   3 
4 

22 

ss 
ss 

15 
16 

24 
24 

ii 

20 
21 

14 

a 

11 

18 
18 
18 
18 
18 

5 
6 

i         7 
8 
9 

20 
20 
20 
20 
20 

13 
14 
15 
16 
17 

22 
22 
22 

se 
ss 
ss 

17 
18 
19 

25 
25 
25 

and 

ss 

3 

15 

and 

4 

4 
5 

15 
15 
15 

i  i 
(< 

5 
6 

7 

23 
23 

and 

4 
5 

25 

25 

ss 

ss 

6 

7 

(( 

8 

18 

10 

23 

ss 

6 

25 

i  i 

8 

15 

15 

(i 

9 

18 

11 

21   ar 

id   3 

23 

ss 

7 

25 

ss 

9 

15 

<  < 

10 

18 

13 

31 

4 

23 

ss 

8 

25 

ss 

10 

15 

i. 

11 

18 

13 

31 

5 

23 

ss 

9 

25 

ss 

11 

15 

<. 

13 

18 

14 

21 

6 

23 

Ss 

10 

25 

ss 

12 

18 

15 

21 

i 

23 

ss 

11 

25 

ss 

13 

16 

and 

4 

21 

8 

23 

ss 

12 

25 

ss 

14 

16 

<  ( 

5 

19   ar 

id   3 

21 

9 

23 

ss 

13 

25 

ss 

15 

16 

c< 

6 

19 

4 

21 

10 

23 

ss 

14 

25 

s , 

16 

16 

t( 

7 

19 

5 

21 

11 

23 

ss 

15 

25 

ss 

17 

16 

a 

8 

19 

G 

21 

''   12 

23 

ss 

16 

25 
25 
25 

ss 
ss 

18 
19 
20 

16 

(( 

9 

19 

7 

21 

13 

23 

ss 

17 

S( 

16 

i( 

10 

19 

8 

21 

14 

23 

ss 

18 

25 

ss 

31 

16 

a 

11 

19 

9 

21 

15 

23 

ss 

19 

25 

ss 

22 

Remarks.— i.  Frequent  and  thorough  use  of  the  Subtraction  Table  will  result  in  great 
facility  in  all  operations  in  this  subject,  and  will  also  aid  in  additions  and  rapid  work  in 
-arithmetical  computations  in  general. 

■^.  The  above  table,  like  the  one  in  addition,  is  given  for  the  teacher's  reference,  to  save 
time  and  labor  in  rapid  mental  exercises. 
2 


18  SUBTR  ACTION. 

79.  When  any  Figure  in  the  Minuend  is  Less  than  the  Corresponding  Figure 
in  the  Subtrahend. 

ExAMTLK. — From  435  take  17<). 

oPEii-VTioN.  ExPi-ANATiON. —  It  is  readily  observed  that  the  units  figure  6  of  the 

subtrahend  cannot  be  taken  from  the  corresponding  figure  of  the  minu- 

•ioo  Minuend.         gjjjj  .  therefore  analyze  the  minuend,  and  transform  it  into  4  hundreds, 

176  Subtrahend.    J  tenii,  15  units;  then  from  the  15  units  take  the  6  units  of  the  subtrahend, 

obtaining  9  units  as  a  remainder,  which  write  as  the  units  of  the  result ; 

2o9  Remainder,      iiaving  reduced  one  of  the  tens  of  the  minuend  to  units,  we  have  only 

3  tens  remaining  in  the  tens'  column  of  the  minuend,  and  since  this  is 

numerically  less  than  the  tens'  figure  in  the  subtrahend,  transform  as  before,  and  read  the  4 

hundreds  and  2  tens  as  3  hundreds  and  13  tens;  then  taking  the  7  tens  of  the  subtrahend  from 

the  12  tens  thus  produced,  write  the  remaining  5  tens  for  the  second  or  tens'  figure  in  the 

result;  having  taken  1  from  the  hundreds'  column,  we  have  3  remaining  in  that  column,  from 

which  take  the  1  hundred  of  the  subtrahend  7,  obtaining  2  as  the  third  or  hundreds'  figure 

of  the  result.     Thus  we  conclude  that  176  subtracted  from  435,  leaves  a  remainder  of  259. 

Remarks. —  1.  This  process  is  called  "borrowing  tens,"  as  each  left-hand  order  is  tenfold 
greater  than  the  order  at  its  right. 

2.  Having  mastered  the  theory,  the  ordinary  and  most  convenient  method  for  practice  is  to 
leave  the  minuend  figure  in  its  original  form  and,  when  borrowing  is  necessary,  add  1  to  the 
succeeding  subtrahend  figure. 

Again,  apply  tliis  method  to  the  exauii)le; 


OPERATIOX. 

435 

176 


259 


Explanation. — Subtract  fi  from  15  leaving  9,  which  write  in  units'  column; 
(adding  1  to  7)  subtract  8  from  13  leaving  5,  which  write  in  its  column;  (adding 
1  to  1)  subtract  3  from  4  leaving  2,  ■ts'hich  write  in  its  column. 


Rule. — I.  So  write  tJie  niunbers  to  he  subtracted,  that  units  of  tJie  same 
order  stand  in  the  same  vertical  line. 

II.  Begin  at  the  right  and  subtract  each  figure  of  the  subtrahend  from 
the  corresponding  figure  of  the  minuend.  When  it  is  necessary,  trans- 
form,  or  borrow  ten,  and  mentally  add  one  to  the  next  subtrahend  figure. 

III.  Write  results  in  their  proper  order. 

KXAMPLK.S    I'OK   PRACTICK. 


80.     J.  From  1524  take  911. 

2.  From  3128  take  1519. 

3.  From  4055  take  2033. 

4.  From  27410  take  13520. 
r5.  From  80500  take  30500. 
6.  From  123706  take  59341. 


7.  From  520200  take  368977. 

S.  From  80090  take  23084. 

9.  From  3406268  take  1998765. 

10.  From  303005  take  89700. 

n.  From  2046  take  1597. 

13.  From  40509300  take  9619475. 


l-l  In  Germany  there  are  2,436,000  kind  owners,  and  in  France  3,226,000. 
How  many  more  in  France  than  in  (lermany? 

1^.  A  dealer  bought  1,732  sheep  and  .sold  to  A  51,  B  147,  C  34,  I)  1000.  and 
to  E  the  remainder.     How  many  did  E  purchase?  , 


SUBTRACTIOX.  19 

16.  A  farmer  raised  1,130  l)ushels  of  wheat,  958  of  barley,  1,275  of  oats,  and 
1,762  of  corn.  If  lie  keep  for  seed  and  feed,  IIG  bushels  of  wheat,  84  of  barley, 
GOO  of  oats,  and  1,150  of  corn,  how  many  bnshels  of  grain  will  he  have  left  to  sell? 

16.  The  equatorial  diameter  of  the  earth  is  41847194  feet,  and  the  polar 
diameter  41707308  feet.  How  many  feet  greater  is  the  equatorial  than  the  polar 
diameter  ? 

17.  If  the  sailing  distance  from  New  York  to  Queenstown  be  2890  miles,  iiow 
far  from  the  latter  port  Avill  a  steamer  be  after  running  1290  miles  from  tiie  port 
of  Xew  York  ? 

18.  Texas  contains  274356  square  miles  and  New  York  47156  square  miles. 
How  many  times  may  the  area  of  New  York  be  taken  from  the  area  of  Texas 
and  Avhat  number  of  square  miles  will  remain  ? 

19.  The  area  of  Brazil  is  3950000  square  miles  and  of  the  United  States 
302G504  square  miles.  How  many  S(iuare  miles  greater  is  Brazil  than  the  United 
States  ? 

20.  A  man  bought  a  farm  for  3250  dollars.  He  built  a  house  on  it  at  a  cost 
of  3850  dollars,  fences  costing  416  dollars  and  then  sold  it  for  7500  dollars. 
What  was  his  gain  ? 

21.  I  bouglit  23,240  acres  of  Dakota  land,  and  sold  at  times  1000,  320,  520, 
640,  3200,  2520,  100,  and  1920  acres.     How  many  acres  had  I  remaining  .^ 

22.  During  a  five  years'  partnership  a  firm  gained  ^123,475.  If  the  gain  the 
first  year  was  $11,425  ;  the  second,  $9,500  ;  tlie  third  as  much  as  the  first  and 
second,  less  $1,120;  the  fourth  equal  to  the  second  and  third;  how  much  must 
have  been  gained  the  fifth  year? 

23.  The  cost  of  my  lot  was  $1,750.  I  paid  for  mason  work  on  my  house, 
$1,210;  for  carpenter  work,  $5,145;  for  plumbing,  $985;  for  decorating,  $1,650; 
for  painting,  $625  ;  for  grading,  sodding,  and  fencing  grounds,  $590.  The 
interest  on  outlays  to  date  of  sale  was  $315.  I  then  sold  the  property  at  a  loss 
of  $20,  receiving  cash  $6,000,  and  a  note  for  the  remainder.  What  was  the  face 
of  the  note? 

24.  My  book-keepers  salary  is  $1,450  per  year.  If  he  requires  for  liis  rent, 
$365;  for  personal  expenses,  $170;  and  for  support  of  his  family,  $775;  what 
amount  will  he  have  left  at  the  end  of  the  year? 

25.  A  Boston  bicyclist  journeying  to  Sun  Francisco,  distant  3,432  miles,  ran 
the  first  week  of  six  days,  an  average  of  77  miles  per  day;  the  €econd  week, 
92  miles  ;  the  third,  84  miles  ;  the  fourth,  106  miles;  the  fifth,  95  miles,  and 
reached  his  destination  at  the  end  of  the  sixth  week.  How  many  miles  did  lie 
run  the  last  week? 


20  MULTIPLICATION. 


MULTIPLICATION. 

81.  Multiplication  is  the  process  of  taking  one  of  two  numoers  as  many 
times  as  there  are  units  in  the  other. 

8*2.  One  of  the  numbers  is  called  the  Multiplicand  and  the  other  the  Multiplier. 
The  numbers  are  also  called  Factors  of  the  product. 

83.  The  Multiplicand  is  the  factor  multiplied. 

84.  The  Multiplier  is  the  factor  by  which  the  multiplicand  is  multiplied. 

85.  The  Product  is  the  resdlt  obtained  by  multiplying  one  number  by 
another. 

86.  The  Factors  of  a  number  are  such  numbers  as  will,  when  multiplied 
together,  produce  tlie  given  number. 

87.  A  Continued  Product  is  the  result  obtained  by  multiplying  several 
factors  together. 

88.  The  Sign  of  Multiplication  is  an  oblique  cross,  x  .  It  is  read 
^'  times,"  or  ''  multiplied  by,''  and  indicates  that  the  numbers  between  which  it 
is  placed  are  to  be  multiplied  together,  or  their  product  obtained.  Thus,  5x2 
is  read  5  times  2,  or,  5  multiplied  by  2. 

Remarks. — 1.  In  practice,  the  multiplier  is  regarded  as  an  abstract  number  and  the  multipli- 
cand as  a  concrete  number  ;  but  as  the  resulting  product  is  the  same  whichever  factor  is  used 
as  a  multiplier,  the  above  relation  is  recognized  only  in  explanations  of  work  done. 

2.  Where  the  multiplicand  is  concrete,  the  product  will  be  concrete  and  of  the  same 
denomination  as  the  multiplicand. 

89.  Multiplication  is  a  short  method  of  performing  addition,  and  like  addi- 
tion may  be  proved  by  subtraction.  Thus,  2  x  2  =  -4  :  that  is,  two  taken  twice 
as  a  factor  =  -i,  or,  2  added  to  2  =  4.  We  prove  this  by  subtraction,  2  from  4 
leaves  2. 

Again,  6  X  7  =  42:  that  is,  6  taken  seven  times  as  a  factor  =  42,  or,  seven  6'8 
added  =  42  ;  this  may  be  proved  by  subtracting  seven  6's  in  succession  from  42, 
when  nothing  remains. 

90.  General  Rules. — l.  If  the  multiplicand  and  multiplier  he  given, 
tlie  product  may  be  found  hij  multiplying  those  factors  together. 

2.  If  the  product  and  multiplier  he  given,  the  multiplicand  may  he 
found  hy  dividing  the  product  by  the  multiplier. 

3.  If  the  product  and  lyiidtiplicand  he  given,  the  multiplier  may  he 
found  by  dividing  the  product  hy  the  midtiplicand. 

4-  If  the  product  of  two  numbers  and  one  of  the  nurnhers  he  given,  the 
other  may  be  found  by  dividing  the  product  by  the  number  given. 


MULTIPLICATION. 


21 


Multiplication   Table. 


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Note. — It  will  be  of  great  advantage  to  tlie  student  to  fully  master  the  above  table.  Any 
delay  caused  by  following  this  suggestion  will  be  offset  by  time  gained  in  subsequent  work  ; 
such  mastery  will  so  increase  the  rapidity  of  work  in  business  applications  as  to  greatly  lessen 
the  labor  of  accounting. 


22 


MCLTIPLICATIOX. 


91.  Multiply 

j'^.vAjni'i.fcs  ft 

IK  ilK. 

\L  ]>KILI.. 

1.     42  bv  3. 

11. 

102  by  4. 

21. 

144  by  1(1. 

SI. 

595  by  13. 

2.     31  by  4. 

n. 

511  by  8. 

22. 

52  by"  11. 

S2. 

70  by  22. 

S.     2:  by  2. 

IS. 

125  by  6. 

23. 

45  by  13. 

SS. 

90  by  25. 

J^.     60  by  5. 

U- 

340  by  2. 

24. 

201  by  15. 

SJ,. 

150  by  23 

5.     51  by  4.  ^ 

lo. 

416  by  3. 

25. 

65  by*20. 

So. 

118  by  11 

6.     75  by  6. 

m. 

99  by  7. 

26. 

411  by  IT. 

S6. 

906  by  15 

7.  91  by  2. 

11. 

133  by  9. 

27. 

932  by  12. 

37. 

450  by  19 

8.     29  by  3. 

IS. 

208  l/v  4. 

28. 

43  by' 19. 

38. 

375  by  14 

9.     57  by  2. 

19. 

666  by  5. 

29. 

Ill  by  23. 

39. 

250  by  18 

to.     95  by  5. 

20. 

89  by  8. 

SO. 

207  by  22. 

40. 

789  by  11 

417 
5 

2085 


9*2.  "When  either  Factor  is  a  Number  within  one's  thorough  knowledge  of 
the  Multiplication  Table. 

Example. — 1.  Multiply  417  by  5. 

OPERATION.  Explanation. — "Write  the  multiplier  5  below  the  unit  figure  of  the  multi- 
plicand, and  multiph"  each  figure  of  the  multiplicand  by  the  multiplier,  thus; 
5  times  7  =  35,  or,  3  tens  -f-  5  units ;  ■write  the  5  imits  in  units'  place  and 
reserve  the  3  tens  to  add  to  the  next  product ;  next,  5  times  1  ten  are  5  tens, 
and  addmg  the  3  tens  reserved  gives  8  tens,  "which  -write  in  tens'  place  ;  next, 
5  times  4  hundreds  are  20  hundreds,  or  2  thousands  ;  write  a  naught,  or  cipher, 

in  the  hundreds'  place  and  the  2  iu  the  thousands'  place,  thus  completing  the  multiplication 

and  obtaining  2085  as  the  product  of  417  multiplied  by  5. 

Example.—,?.  Multiply  123  by  12. 

Explanation. — Multiply  each  figure  of  the  multiplicand  by  the  multiplier, 
12  ;  12  times  3  =  36,  or,  3  tens -j- 6  units;  write  the  6  in  units'  place  and  reserve 
the  3  tens  to  add  to  the  next  product ;  next,  12  times  2  tens  are  24  tens,  and 
adding  the  3  tens  reserved  gives  27  tens,  or  2  hundreds-)-  7  tens;  write  the  7  in 
tens'  place  and  reserve  the  2  hundreds  to  add  to  the  next  product;  next,  12  times 
1  hundred  are  12  hundreds,  and  adding  the  2  hundreds  reserved  gives  14  hun- 
dreds, or,  1  thousand  4  hundreds,  which  write  in  hundreds'  and  thousands'  places,  thus  com- 
pleting the  multiplication  and  obtaining  1476  as  the  product  of  123  multiplied  by  12. 

Rule. —  Write  the  factors  one  below  the  other,  arranged  so  that  figures 
of  like  orders  will  stand  in  the  same  vertical  line.  Multiply  each  figure 
of  the  upper  factor,  heginjiing  at  the  right,  hy  the  lower  factor,  placing 
in  order  the  last  figure  of  the  product  so  obtained,  and  carrying  to  tlie 
next  product  all  figures  except  the  last ;  continue  so  doing  until  the  last 
product  is  found,  which  u-riie  in  full. 


0PER.\T10N. 

123 
12 

1476 


KXA>II'M.N    >oK    >IKNTAI.   I'ltACTICK. 


4- 


6. 


93.  Multiply 

201  by  8. 

1325  by  2. 

IS. 

hll  by  16. 

li>. 

641  l)y  13. 

507  by  5. 

8. 

2108  by  li. 

14. 

1603  by  9. 

20. 

7122  i)y  5. 

1001  by  12. 

9. 

511  by' 15. 

lo. 

3006  by  14. 

21. 

984  by  8. 

311  by"  6. 

10. 

293  b"y  12. 

10. 

249  by  7. 

22 

2260  by  12 

805  by  9. 

11. 

1801  by  13. 

17. 

519  by  8. 

23. 

461  l)y"l4. 

1203  by  8. 

12. 

684  by' 14. 

18. 

1122  in-  11. 

2J,. 

3542  in-  15 

MULTIPLICATIOK. 


•->3 


EXAMPLKS   FOK   WRITTKX   I'KACTICK. 


94.  Multiply 

1. 

2168  by  9. 

€,. 

35G142  by  18. 

JL 

o 

31046  by  16. 

t . 

2147603  by  8. 

12. 

3. 

599  by  12. 

S. 

15286097  by  15. 

13 

Jf- 

2170  by  13. 

0. 

508093240  by  13. 

n 

5. 

50890  bv  11. 

JO. 

6381201432  i)y  14. 

]■', 

99084160024  by  15. 
294640205580  by  9. 
6620.5380777  by  7. 
897352468004  by  12. 
21430206041  bv  15. 


Multiplicand. 
Multiplier. 

Units. 
Tens. 
Hundreds.    1251 


417 
352 


834 
2085 


95.     When  the  Multiplier  consists  of  two  or  more  figures. 
Example. — Multiiily  417  l»y  352. 

OPERATION.  Ex  PI,  AX  AT  I  OX. — Write  the  numbers  one  below  the  other  in  the 

00  same  unit  order  from  the  right.    Then,  beginning  with  the  unit  figure 

t  .  of  the  lower  factor  multipl}-;  2  times  7  units  are  14  units  =  1  ten  -f 
5c~  4  units  ;  write  the  4  units  in  units'  column  and  add  the  1  ten  to  the 
next  product;  next,  2  times  1  ten  are  2  tens  and  the  1  ten  added  makes 
3  tens,  which  write  in  tens'  place  ;  next,  2  times  4  hundreds  are  8 
hundreds,  which  write  in  hundreds'  place,  giving  834  as  the  first  partial 
product,  or  the  product  of  the  upper  factor  multiplied  by  the  unit 
figure  of  the  lower  factor.  Next  take  the  tens'  figure  of  the  lower 
number  as  a  multiplier;  7  taken  5  tens  or  50  times  =  35  tens,  or  350  ; 
write  the  5  of  the  number  35  in  tens'  column,  or  below  the  8  tens  of 
the  first  partial  product,  and  carry  the  3  of  the  35  to  the  next  product; 
next,  5  times  1  are  5  and  the  3  to  carry  added  make  8,  which  write 
under  the  8  of  the  first  partial  product ;  then,  5  times  4  are  20,  which 
write  still  to  the  left,  making  the  second  partial  product  2085  tens.  Next,  take  the  third  figure, 
or  hundreds,  of  the  lower  factor,  as  a  multiplier;  3  times  7  hundreds  are  21  hundreds  ;  write 
the  1  in  the  hundreds'  place  and  reserve  the  2  for  the  next  product  ;  then,  3  times  1  are  3  and 
2  to  carry  makes  5,  which  write  in  its  order  ;  then,  3  times  4  are  12,  which  write  still  to  the 
left,  having  as  a  product  1251.  Since,  however,  the  several  figures  of  the  factor  taken  as  a 
multiplier  were  of  successive  orders  of  units. 

The  first  partial  product  834  =  834  simple  units. 

The  second  partial  product  2085  =  2085  tens  =         20850      " 
The  third  partial  product  1251  =  1251  hundreds  =  125100      " 


146784 


And  the  sum  =  146784 
Therefore,  146784  is  found  to  be  the  product  of  the  numbers  417  and  352. 

Rule. — I.  Place  the  multiplier  helow  the  multiplicand,  the  unit  figures 
in  the  same  vertical  line. 

II-  Beginning  with  the  unit  figure,  multiply  all  the  figures  of  the  mul- 
tiplicand bij  each  successive  figure  of  the  multiplier,  writing  the  first 
figure  obtained  in  each  partial  product  directly  below  the  figure  by  which 
it  was  multiplied.    Add  the  partial  products. 

Remark. — The  object  of  writing  each  succeeding  partial  product  below  and  one  place  to 
the  left  of  its  predecessor,  is  that  imits  of  the  same  grade,  or  order,  ma}%  for  convenience  in 
adding,  be  found  in  tlie  same  vertical  line;  this  arrangement  precludes  the  necessity  of  filling 
the  vacant  orders  witli  ciphers. 

As  before  shown,  the  arrungemeiit  of  factors  will  not  vary  the  result ;  as, 
4  X  5  =  20,  also  5  X  4  =  20  ;  therefore,  in  business  or  school  ])ractice,  arrange 
factors  in  such  order  as  to  save  time  and  space;  by  so  doing,  problems  otherwise 
long  and  difficult,  may  be- solved  Ijy  mental  ])rocesses. 


24  MULTIPLICATION. 

Example.— Multiply  120000  by  7256. 

Explanation.  —  Consider  the  factors  as  reversed  in  order ;  thus,  7256  X  120000.  Theo 
multiply  the  7256  mentally  by  12,  and  to  the  product,  87072,  annex  four  ciphers,  because  the 
12  was  not  12  simple  units  but  12  units  of  the  tifth  order,  or  tens  of  thousands. 

9().  When  one  Factor  is  10,  100,  1000,  10000,  or  1  with  any  number  of 
ciphers  annexed. 

E.\AMPLE.— Multiply  324  by  lOUO. 

ExPL.\NATiON. — Since  there  are  three  ciphers  in  tlie  multiplier,  annex  three  to  the  multipli- 
cand, 324,  thus  obtaining  the  product,  324000. 

Rule. — To  the  one  factor  annex  as  many  ciphers  as  there  are  ciphers 
in  the  other  factor. 

97.  A  Composite  Nuinbei*  is  a  number  that  may  be  resolved  or  separated 
into  integral  factors  ;  or,  it  is  a  number  that  may  be  formed  by  multiplying 
together  two  or  more  numbers;  thus,  12  =  4  X  3  ;  or  12  =  2  x  2  X  3  ;  or  4  X  3 
=  12;  or  2  X  2  X  3  =  12. 

98.  When  the  Multiplier  is  a  Composite  Number. 

Whenever  it  is  required  to  find  the  product  of  numbers  one  or  more  of  which 
is  composite,  the  result  may  be  obtained  by  using  as  multipliers  the  factors  of  l 
such  composite  number  or  numbers;  thus,  6  X  4  =  24,  or  6  x  (2  X  2)  =  24. 

Rule. — Separate  the  multiplier  into  its  factors.  Multiply  the  miMipli- 
cand  by  one  of  these  factors,  that  product  hy  another  factor,  and  so  on, 
using  in  succession  all  the  factors;  the  last  product  ivlll  be  the  result 
required. 

Rem.ark. —  Since  the  order  in  which  factors  are  used  will  not  vary  the  product,  the  student 
is  recommended  to  seek  the  simplest  number — the  one  most  easily  factored  —  as  a  multiplier. 

KXAMPLES  rOK   PRACTICE. 

99.  1.  Multijily  41  by  15,  itsing  as  factors  3  and  5, 
2.  Multiply  17  by  21,  using  as  factors  7  and  3. 

'3.  ;Multi])ly  111  by  24,  using  as  factors  3,  2,  and  4 

4.  Multiply  1157  by  30,  using  as  factors  6  and  6. 

o.  Multiply  2019  by  45,  using  as  factors  5,  3,  and  3. 

6.  Multiply  8T002  by  9G,  using  as  factors  6,  4,  and  4. 

7.  Multiply  54235  by  144,  using  as  factors  12  and  12. 

8.  Multiply  54235  by  144,  using  as  factors  9  and  16. 

9.  Multiply  54235  l)y  144,  using  as  factors  9,  4,  and  4. 

10.  Multiply  54235  by  144,  using  as  factors  3,  3,  2,  and  8. 

11.  ^Multiply  54235  by  144,  using  as  factors  3,  3,  2,  2,  and  4. 

12.  Multiply  54235  by  144,  using  as  factors  3,  3,  2,  2,  2,  and  2. 

13.  Multiply  81  by  64,  using  as  factors  8  and  8, 

14.  Multiply  04  Ijy  81,  using  as  factors  9  and  9. 

lo.  Multiply  81  by  04,  using  as  factors  of  81.  0  au'I  9,  and  as  factors  of 
64,  8  and  8. 


100.  Multiply 

1.     1431  by  7000. 

e 

^.  900  by  2104G. 

7 

S.     1969  by  54  =  9  x  6. 

8 

Jt.     171548  by  1500  =  15  X  100. 

9 

5.     1653  by  25000  =  5  X  5  X  1000. 

10 

MULTIPLICATION-.  25 

KXAMPtKS   COMBINING   EtEMENTAKY   PKINCIPLES   PKEVIOUSI.Y   EXPLAINED. 

3500  by  72  =  6  X  12. 
1921  by  450  =  9  X  5  X  10. 
321058  by  144  =  12  x  12. 
504  by  288  =  9  X  8  X  4. 
1043  by  105  =  7x3x5. 

11.  A  clerk  sold  9  shirts  ut  80  cents  each,  2  neck-ties  at  35  cents  each,  10  col- 
lars at  25  cents  each,  a  pair  of  gloves  for  75  cents,  and  two  suits  of  underwear 
at  95  cents  ]}qv  suit.     What  was  the  price  of  all  ? 

12.  I  bought  15  cows  at  32  dollars  per  head,  a  pair  of  horses  for  245  dollars, 
a  harness  for  22  dollars,  and  81  sheep  at  3  dollars  per  head  ;  what  was  the  total 
cost  of  my  purchases? 

13.  The  cost  of  furnishing  a  house  was,  for  parlor  and  library  furniture  762 
dollars,  halls  150  dollars,  dining  room  and  kitchen  295  dollars,  chambers  648 
dollars,  stoves  and  furnace  350  dollars,  carpets  and  curtains  825  dollars,  what 
was  the  total  cost  ? 

IJf.  14250  dollars  was  paid  for  four  houses,  the  first  costing  2750  dollars,  the 
second  400  dollars  more  than  the  first,  the  third  250  dollars  less  than  the  first 
and  second  together,  and  the  fourth  the  remainder.  Find  the  cost  of  the  fourth 
house  ? 

15.  Find  the  difference  between  the  continued  products  of  91  X  4  X  3x11x9 
and  5x5x12x4x6x7. 

16.  Find  the  difference  between  seven  units  of  the  sixth  order  and  the  con- 
tinued product  of  15x6x5x12x4x7x11x8x2x9. 

17.  A  merchant  having  17462  dollars  to  his  credit  in  a  bank,  gave  checks  as 
follows:  for  dry  goods  5416  dollars,  groceries  5995  dollars,  boots  and  shoes  1416 
dollars,  hardware  1850  dollars,  and  drew  out  500  dollars  for  family  expenses;  what 
amount  was  left  in  the  bank  ? 

18.  Exchanged  a  city  block  valued  at  35000  dollars,  for  a  farm  of  175  acres 
valued  at  95  dollars  per  acre,  eight  horses  at  110  dollars  cacii,  14  cows  at  28 
dollars  each,  225  sheep  at  4  dollars  each,  farm  machinery  valued  at  825  dollars, 
and  received  the  balance  in  cash.     How  much  cash  was  received  ? 

19.  A  drover  bought  135  horses  at  an  average  j^rice  of  115  dollars  for  100  of 
them,  and  125  dollars  per  head  for  the  remainder  ;  he  sold  25  at  100  dollars  per 
head,  twice  that  number  at  twice  the  price  per  head,  and  the  remainder  at  67 
dollars  per  head.     IIow  much  was  gained  or  lost  ? 

20.  A  ranchman  sold  to  a  trader,  46  ponies  at  60  dollars  i)er  j)air,  116  calves 
at  9  dollars  per  head,  41  cows  at  35  dollars  per  head,  and  a  pair  of  mules  for  375 
dollars.  He  received  in  part  payment,  15  barrels  of  flour  at  9  dollars  per  barrel, 
11  hundred  weight  of  bacon  at  12  dollars  i)er  hundred  weight,  4  suits  of  clothes 
at  22  dollars  \k'V  suit,  2  saddles  at  13  dollars  each,  a  wagon  at  75  dollars,  a  set  of 
furniture  for  58  dollars,  and  the  remainder  in  cash.  What  amount  of  cash  did 
the  trader  i)ay  ?  '     .     ' 


26  MULTIPLICATION. 

MISCELLANEOUS   EXAMPLES. 

101.  1.  The  United  States  export  105,000  sewing  machines  yearly.  If  each 
machine  does  the  work  of  12  women,  what  is  the  value  of  the  labor  thus  con- 
tributed by  the  United  States  to  other  nations  each  year  of  306  working  days,  if 
labor  be  estimated  at  $1  per  day? 

2.  The  Union  Pacific  Railway  is  1777  miles  in  lengtli,  and  was  built  at  an 
average  cost  of  $106,775  per  mile;  what  was  the  total  cost  of  construction  ? 

3.  The  bills  issued  by  the  U.  S.  Treasury  for  National  Bank  circulation,  are 
in  denominations  of  ^1,  §2,  $5,  810,  $20,  $50,  $100,  $500,  and  $1000.  How  much 
money  has  one  possessing  73  bills  of  each  denomination  ? 

J^.  The  gold  coins  of  the  U.  S.  are  in  denominations  of  $1,  $2.50,  $3,  $5,  $10, 
and  $20.     How  much  money  in  a  bag  containing  365  of  each  of  these  coins  ? 

5.  The  U.  S.  notes — greenbacks  —  are  of  the  following  denominations,  viz.: 
$1,  $2,  $5,  $10,  $20,  $50,  $100,  8500,  $1000,  $5000,  and  $10000.  How  large  a 
debt  could  be  paid  with  7  of  each  of  the  above-named  greenbacks  ? 

6.  How  many  feet  of  wire  will  be  required  to  fence  a  field  11 10  ft.  square,  with 
six  wires  on  each  of  the  four  sides  ? 

7.  What  is  the  amount  of  the  following  bill  ? 

28  lb.  Lard  @  l\f  per  lb.  110  lb.  Beef  @  14^'  per  lb. 
46  bii.  Salt  @  15^'  per  bu.  50    "    Butter  @  32^  per  lb. 

17    "    Apples  @  45^  per  bu.  4  pk.  Onions  @,  35^  per  pk. 

61  lb.  Pork  @  9/  per  lb.  15  bu.  Potatoes  @  75^'  per  bu. 

8.  Find  the  total  cost : 

4  cd.  Hard  Wood  @  $6  per  cord.  13  tons  Furnace  Coal  @  $5  i)er  ton. 

11    "    Soft  Wood  @  $3  per  cord.  7-    "     Stove  Coal  @  $6  i)er  tun. 

9  loads  Kindling  @  $2  \^cv  load.  2     "     Cannel  Coal  @  $9  per  ton. 

9.  Find  the  cost  of 

7  lb.  Tea  @  65^  per  lb.  9  lb.  Java  Coffee  @  31^  per  lb. 

50    "    A  Sugar  @  7^-  per  lb.  52    "    Br.  Sugar  @  bf-  per  lb. 

15    ''   Cheese  @  13^-  per  lb.  25    "   C  Sugar  @  6^-  per  lb. 

10.  What  must  be  paid  fur  llie  following  goods  ? 

7  yd.  Prints  @  7^  per  yd.  11  yd.  Jeans  @  !%(/:  per  yd. 

61  "    Sheeting  @  13^  per  yd.  29    "    Calico  @  ^<P  per  yd. 

77  "    Ticking  @  15^  \)vv  yd.  14    "    Delaine  @  23^  per  yd. 

17  "    Drilling  @  16^'  per  yd.  25    "    Gingham  @  12^  per  yd. 

11.  Find  the  total  cost : 

67  yd.  Moquette  Carpet  %  $3  per  yd.  32  yd.  Border  No.  1  @  $3  per  yd. 
131  "'  Brussels  "  @  $2  i)er  vd.  70  "  "  "  2  @  $2  per  yd. 
100    ■•    Ingrain  "       @  $1  per  yd.      45    '•         "         "    3  @  $1  per  yd. 

13.   The  Boston  '•boot-maker"  will  enable  a  workman  to  make  300  i)airs  of 
boots  daily.      How  many  i»airs  can  he  make  in  a  year  having  309  working  days? 


MULTIPLICATIOX.  27 

13.  My  grain  sales  for  the  year  1888  were  as  follows  : 

516  bu.  White  Wheat  @  85^/  per  bu.  250  bu.  Peas  @  95^'  per  bu. 

723    "    Rod           "      @  95^-  per  bu.  287  ''    Rye  @;  92^/-  per  bu. 

941    "    Barley  @  73^  per  bu.  635  "'•    Beans  @  75^-  per  bu. 

1625    "    Oats  @  32^  per  bu.  321  ''    Buckwheat  @  85'/ j.or  Im. 
How  much  was  received  for  all  ? 

14.  In  New  York  State  a  bushel  of  barley  weighs  48  lb.,  of  clover  seed  6(t  lb., 
of  flax  seed  55  lb.,  of  beans  62  lb.,  of  buckwheat  48  lb.,  of  rye  56  11).,  of  corn 
58  lb.,  of  oats  32  lb.,  of  potatoes  GO  lb.,  of  timothy  seed  44  lb.,  and  of  wheat 

60  lb.     What  will  be  the  total  weight  of  5  bushels  of  each  of  the  products  named  ? 

15.  In  freighting,  lime  and  flour  are  each  estimated  to  weigh  200  lb.  per  barrel; 
pork  and  beef  each  320  lb.;  apples  and  potatoes  150  lb.  each;  cider,  whiskv,  and 
vinegar  each  350  lb.  What  will  be  the  freight  at  20^'  per  liundred  jtounds,  on  a 
car  containing  15  barrels  of  each  of  these  pVoducts  ? 

16.  I  bought  10  acres  of  land  at  $2250  per  acre  and  laid  it  out  in  75  city  lots, 
expending  $4725  for  grading  and  streets,  $680  for  sidewalks,  and  $87  for  orna- 
mental trees.  I  then  sold  40  of  my  lots  at  $500  each,  20  at  $450  each,  and 
exchanged  the  remainder  for  a  farm  of  110  acres,  the  cash  value  of  which  was 
$65  per  acre.     How  much  Avas  gained  or  lo.<t  ? 

17.  A  gardener  rented  5  acres  of  land  for  $20  per  acre  and  paid  $63  for  seeds, 
$20  for  fertilizers,  $246  for  labor,  and  $52  for  freight.  He  sold  2145  bushels  of 
turnips  for  $429,  1710  bushels  of  beets  for  $513,  4350  bunches  celery  for  $174, 
and  800  heads  cabbage  for  $40.     What  was  his  gain? 

18.  A  man  earning  $2.50  per  day,  works  306  days  per  year  for  five  years.  His 
annual  expenses  are,  for  board  $156,  for  clothing  $47,  for  charity  $12,  and  he 
expends  $2  per  week  for  incidentals.  If  he  deposit  his  surplus  each  year  in  a 
Savings  Bank,  what  amount  will  he  deposit  during  the  tinu'? 

19.  The  U,  S,  coupon  bonds  are  in  denominations  of  $50,  $100,  $500,  and 
$1000,  and  the  registered  bonds  in  denominations  of  $50,  $100,  $500,  $1000, 
$5000,  and  $10000,  01  the  4J-'s  of  1891,  and  the  4's  of  1907,  there  are  registered 
bonds  of  the  denominations  of  $20,000  and  $50,000.  What  would  be  the  ajrffre- 
gate  face  value  of  twelve  of  each  of  the  bonds  above  named  ? 

20.  A  man  rented  a  farm  of  132  acres  of  grain  land,  67  acres  of  i)asture  land, 
and  45  acres  of  meadow  land  ;  paying  for  the  grain  land  $7  per  acre,  for  the 
pasture  land  $4  per  acre,  and  for  the  meadow  land  $11  per  acre.    'He  produced 

61  bushels  of  oats  per  acre  on  45  acres,  32  bu.  barley  per  acre  on  30  acres,  75  bu. 
corn  per  acre  on  15  acres,  150  bu.  potatoes  on  9  acres,  28  bu.  buckwheat  on  20 
acres,  and  24  bu.  beans  per  acre  on  tiie  remainder  of  the  grain  land.  He  re-let 
the  pasture  land  for  $200,  and  on  tlic  meadows  cut  2  tons  per  acre  of  hay  worth 
$13  per  ton.  If  he  paid  $695  for  labor  and  $467  for  other  expenses,  did  he  gain 
or  lose,  estimating  oats  at  $275,  barley  at  $672,  corn  at  $394,  potatoes  at  $743, 
buckwheat  at  $420,  and  beans  at  $2  per  bushel  ? 


28  DIVISION. 


DIVISION. 

102.  Diyision  is  the  process  of  finding  how  many  times  one  number  is 
contained  in  another  of  the  same  kind. 

103.  The  Dividend  is  the  number  divided. 

104.  The  Divisor  is  the  number  by  which  the  dividend  is  divided. 

105.  The  Remainder  is  the  part  remaining  when  the  division  is  not  exact. 

106.  The  Sign  of  Division  is  the  character  -=-  ;  it  indicates  that  the  num- 
ber before  it  is  to  be  divided  by  tlie  number  after  it.  Thus,  24  -^  3  =  8,  is  read 
24  divided  by  3  equals  8.  We  see  by  this  operation  that  3  is  an  exact  divisor 
of  24,  also  that  3  and  8  are  factors  of  24. 

Remark. — From  the  above  it  is  clear  that  the  dividend  in  division  corresponds  to  the  product 
in  multiplication,  and  the  divisor  and  quotient  to  the  multiplier  and  multiplicand,  or  the  factors 
in  multiplication. 

107.  General  Principles.  —  l.  Multiplying  the  dividend  multiplies  the 
quotient.      Thus,  Jf8  ^  6  =  8;  {J^S  x  2)  -i-  6  =  16. 

2.  Dividing  the  divisor  multiplies  the  quotient.  Thus,  4^  -i-  6  =  8;  J!^8  -^ 
{6-^2)  =Jt8  -^3=  16. 

3.  Dividijig  the  dividend  divides  the  quotient.  Thus,  4.8  -i-  6  =  8;  (4<§  -i-  2) 
~6  =  24  -^  6'  =  4. 

U-  Multiplying  the  divisor  divides  the  quotient,  lints,  48  ^  6  =  8;  48  -i- 
{6  X  2)  =  48  -^  12  =  4. 

108.  General  Law. — I.  Any  change  hi  the  dividend  produces  a  like  change 
in  the  quotient. 

II.  Any  change  in  the  divisor  produces  an  opposite  change  in  the  quotient. 
III.  A  like  change  in  both  dividend  and  divisor  will  not  change  the  quotient. 

109.  General  Rnles.— i.  //  the  dividend  cmd  divisor  be  given,  the 
quotient  may  be  found  by  dividing  the  dividend  by  the  divisor. 

2.  If  the  dividend  and  quotient  be  given,  the  divisor  may  be  found  by 
dividing  the  dividend  by  the  quotient. 

3.  If  the  divisor  and  quotient  be  given,  the  dividend  may  be  found  by 
multiplying  the  divisor  by  the  quotient. 

4.  If  the  divisor,  quotient,  and  remainder  be  given,  the  dividend  may  be 
found  by  multiplying  the  divisor  by  the  quotient  and  adding  the  remain- 
der to  the  product. 


DIVISION. 


29 


110.  To  ProYe  Division. — Divide  the  dividend  hy  the  quotient,  or  mul- 
tiply the  divisor  hy  the  quotient.  In  divisions  which  are  not  exact,  add  the 
remainder  to  the  iiroduct  of  the  divisor  and  quotient;  the  sum  thus  obtained 
should  he  the  dividend. 

111.  The  Reciprocal  of  a  number  is  one,  or  unity,  divided  by  that  number. 
A  reciprocal  will  be  produced  by  changing  the  relation  of  dividend  and  divisor; 

as,  28  -^  4  =  7,  while  4  -4-  28  =  i  ;  the  resulting  \  is  the  reciprocal  of  the  first 
quotient  7. 

MENTAL,   EXERCISES. 

112.  What  is  the  quotient  of 

1.  16  -T-  2,  4,  8. 

2.  20  -T-  2,  4,  5,  10. 
4,  8,  2,   7,  14. 


•  •4. 
5. 

6. 
7. 
8. 
9. 
10. 


90 


11. 
12. 

13. 

14- 
15. 
16. 
11. 
18. 
19. 
20. 


125  ^  5,  25. 

48  ^  4,  12,  3,  6,  2. 

64  -^  8,  4,  32,  2,  16. 

120  H-  20,  3,  8,  5,  12. 

80  -4-  4,  16,  10,  20,  8. 

144  -^  12,  8,  6,  4,  3,  24. 

175  ^  35,  7,  5. 

96  H-  6,  8,  32,  12,  16. 

108  --  3,  2,  9,  6,  12,  27. 

200  -f-  5,  10,  20,  8,  4. 


3,  6,  15,  9. 
45  H-  9,  15,  5,  3. 

36  -^  4,  18,  12,  2,  9. 

6,  2,  12,  24. 
84  --  7,  4,  2,  12,  21. 
100  -^  5,  25,  2,  4,   10. 
24  -^  6,  2,  4,  12,  8. 

Operations  in  Division  are  of  two  kinds ;   Short  Division  and  Long  Division. 

113.  In  Short  Division,  operations  are  restricted  to  those  divisions  in 
which  the  divisor  consists  of  one  figure,  or  is  a  number  coming  within  one's 
thorough  knowledge  of  the  multiplication  table. 

114.  When  the  Divisor  consists  of  only  one  figure. 

Example.— Divide  6482  by  2. 

Explanation. — Write  the  divisor  at  the  left  of  the  dividend,  sepa- 
rating them  by  a  line,  next  draw  a  line  below  the  dividend  and  then 
divide  each  figure  of  the  dividend  by  the  divisor,  writing  the  quotient 
below  the  figure  divided.  Thus,  2  is  contained  in  6  thousands,  3  thou- 
sands times  ;  write  the  3  below  the  6  in  thousands'  column  ,  next,  2  is 
contained  in  4  hundreds,  2  hundreds  times ;  place  the  2  below  the  4  in 

hundreds'  column  ;  2  is  contained  in  8  tens,  4  tens  times  ;  write  the  quotient  in  tens'  column  ; 

2  is  contained  in  2  units,  1  unit  times,  or  once  ;  write  1  in  units'  place,  thus  completing  the 

division,  and  obtaining  3241  as  a  quotieiit. 

115.  When  the  Divisor  is  a  Number  within  one's  thorough  knowledge  of 
the  Multiplication  Table. 

Example.— Divide  31605  by  15. 

Explanation. — Write  the  terms  as  before.  Divide  31  by  15  and  obtain 
2,  which  write  below  the  1  as  the  first  figure  of  the  quotient ;  next,  15  is 
contained  in  16,  once ;  write  1  in  hundreds'  column  ;  15  in  10,  0,  or  no 
times  ;  write  the  0,  or  cipher,  in  tens'  column  ;  15  in  105,  7  times  ;  write 
the  7  as  units  of  tlie  quotient,  tlius  completing  the  division,  and  obtaining 
the  quotient  2107. 


OPERATION. 


2  )  6482 
3241 


OPERATION. 


15  )  31605 
2107 


30 


DIVISION. 


116.     When  any  Figure  or  Figures  of  the  Dividend  will  not  Exactly  Contain 
the  Divisor. 

Example. — Divide  394015  bv  8. 


OPEKATIOX. 


8  ) 394015 


49251^ 


Explanation  — "Write  the  terms  as  before.  Since  8  hundreds  of 
thousands  is  not  divisible  by  the  divisor  8,  unite  the  3  hundreds  of  thou- 
sands and  the  9  tens  of  thousands,  obtaining  39  tens  of  thousands  ;  divide 
this  by  8  and  obtain  for  the  first  figure  of  the  quotient  4  tens  of  thousands, 
with  a  remainder  of  7  tens  of  thousands  ;  write  the  4  below  the  9  as  the 
tens  of  thousands  of  the  quotient,  and  unite  the  7  tens  of  thousands  to  the 
4  thousands  of  the  dividend  and  divide ;  8  is  contained  in  74  thousands,  or  7  tens  of  thou- 
sands -|-  4  thousands,  9  thousands  times  witli  a  remainder  of  2  thousands ;  write  the  9  in 
the  column  of  thousands,  and  unite  the  2  thousands  to  the  next  figure  of  the  dividend  '  8  is 
contained  in  20  hundreds,  2  hundreds  times  with  a  remainder  of  4  hundreds ',  write  the  2 
hundreds  in  the  column  of  hundreds,  and  unite  the  4  hundreds  to  the  next  figure  of  the 
dividend  ;  8  is  contained  in  41  tens,  or  4  hundreds  +  1  ten,  5  tens  times,  with  a  remainder  of 
1  ten  ;  write  the  5  in  tens'  column  and  uuite  the  1  ten  to  the  last  figure  of  the  dividend ;  8  is 
contained  in  15  units,  1  unit  times,  or  once,  with  a  remainder  of  7  units,  or  7  ;  write  the 
remainder  over  the  divisor  in  the  form  of  a  fraction  and  annex  the  result  to  the  entire  part  of 
the  quotient,  thus  obtaining  49251|  as  the  complete  quotient  of  394015  divided  by  8. 


Rule. —  I.  Write  the  divisor  at  the  left  of  the  dividend  with  a  line 
separating  them  . 

11.  Beginnifig  at  the  left,  divijde  each  figure  of  the  dividend  by  the 
divisor,  and  write  the  resulting  quotient  underneath  the  dividend. 

in.  If  after  any  division  there  he  a  remainder,  regard  this  remainder 
as  prefixed  to  the  next  figure  of  the  dividend,  and  divide  as  before. 

IV.  Sliould  any  partial  dividend  considered,  be  less  than  the  divisor, 
place  a  cipher  in  the  quotient  and  regard  the  undivided  part  as  prefixed 
to  the  succeeding  figure  in  the  dividend  and>  again  divide. 

y.  //  the  division  is  not  exact,  write  the  remainder  over  the  divisor  in 
fractional  form,  and  annex  the  result  to  the  integral  part  of  the  quotient. 


EXAMPLES  FOR   I'KACTICE. 


117.  Divide 

1. 

646  by  2. 

8. 

143258  by  11. 

lo. 

7600  by  16. 

2. 

945  by  3. 

fK 

81052  by  13. 

16. 

240000  by  13. 

3. 

1124  by  4. 

10. 

5841226  by  14. 

17. 

20416201  by  15 

h. 

2645  by  5. 

11. 

90090  by  7. 

IS. 

952451  by  17.  • 

5. 

31562  by  8. 

12. 

163208  by  15- 

19. 

200468  by  18. 

6. 

60703  by  9. 

13. 

21406  by  8. 

20. 

1119306  by  10. 

7. 

2075  by  12. 

u. 

51007  by  11. 

21. 

8476432  by  12. 

118.     "When  the  Divisor  is  a  Composite  Number. 

When  the  divisor  is  a  composite  number  the  operation  may  be  simplified  by 
usinff  tlie  factors  of  the  divisor. 


DIVISION.  31 

Example. — Divide  15552  bv  288. 


OPERATION. 


Explanation.— First  resolve  the  number  288  into  the  factors 

3  )  15552  '^,  S,  12.    Then  dividing  the  dividend  by  the  factor  3  obtain  5184, 

—  the  first  ciuotient;  dividing  this  quotient,  treated  as  a  new  divi- 

8  )  5184  1st  quotient,      dend,  by  the  factor  8  obtain  648  as  the  second  quotient ;  again, 

..  dividing  by  the  factor  12  obtain  54,  the  third,  or  final  quotient, 

12  )  648  2nd       "  which  is  the  quotient  required.     Hence  14.552  divided  by  288 

„  ,  equals  54. 

04   3rd 

Rule. — Divide  the  dividend  by  any  one  of  the  factors,  and  the  quotient 
thus  obtained  hy  another  of  the  factors,,  and  so  on  until  all  of  the  factors 
have  been  used  as  a  divisor.    The  last  quotient  will  be  the  required  result. 

EXA31PI.KS  FOK   PRACTICE. 

1 19.  1.     Divide  216  by  72,  using  the  factors  8  j^nd  9. 

2.  Divide  1100  by  55,  using  the  factors  5  and  11. 

3.  Divide  5280  by  480,  using  the  factors  4,  12,  and  10. 

4.  Divide  31248  by  144,  using  the  factors  12  and  12. 

5.  Divide  31248  l)y  144,  using  tlio  factors  9  and  16. 

6.  Divide  31248  by  144,  using  the  factors  8  and  18. 

7.  Divide  31248  by  144,  using  the  faptors  8,  2,  and  9. 

8.  Divide  -31248  by  144,  using  the  ftictors  4,  2,  3,  and  6. 

.9.  Divide  31248  by  144,  using  the  fi^ctors  2,  2,  2,  3,  3,  and  2. 

10.  Divide  2025  by  45,  using  the  factors  3  and  15. 

11.  Divide  2025  by  45,  using  the  factors  3,  3,  and  5. 

12.  Divide  2025  by  45,  using  the  factors  9  and  5. 

Remark. — The  pupil  will  observe  that  tlie  order  in  wiiich  the  factors  are  used,  does  not 
vary  the  result. 

120.  To  find  the  True  Remainder  after  Dividing  by  the  Factors  of  a  Com- 
posite Number. 

Example.— Divide  1347  by  105,  using  tlio  factors  5,  3,  and  7. 

OPERATION.  Explanation.— Divide  the  given  dividend  by  3, 

_  .  .  •  obtaining  the  quotient  209,  with  2  units  for  a  remain- 

^  '         '    tniits.  ^gj..  jjjg  quotient  269  is  composed  of  units  equal  in 

3  "i  or-qs's    ,  9        ■±.  value  to  5  times  those  of  the  given  dividend,  and  may 

___  *  be  written  2696'8 ;  thfe  remainder,  2,  is  of  the  same 

7  )  89"^'^  +  2'''''  =    10     "  "°^'  value  as  the  given  dividend,  and  is,  therefore,  a 

part  of  the  O'we  remainder;  next  divide  the  quotient 

13W5's  _,_  5i5'a  _.  -vg     It  26y^'s  by  3  obtaining  89  for  a  quotient  and  2  for  a 

—  remainder.     The  units  of  whicli  tiie  quotient  89  is 

rue  rem.      composed,  are  equal  in  value  to  15  times  those  of 

•'-"To  5  (juotient.  tijg  given  dividend  and  may  be  written  SQi^'s  ;  the 

remainder  is  25's  and  equals  5  X  2,  or  10  units  of  the 
given  dividend,  next  divide  by  7  which  gives  the  quotient  12,  with  5  for  a  remainder;  the 
quotient  12  is  composed  of  units  equal  in  value  to  105  times  those  in  the  given  dividend  and  may 


32  DIVISION. 

be  written  12'"^'^ ;  the  remainder  is  5'^'^  and  equals  15  X  5,  or  75  units  of  the  given  dividend. 
The  sum  of  the  remainders,  2  units,  2^'^  or  10  units,  and  5'^'*,  or  75  units,  equals  87,  the  true 
remainder,  and  the  result  of  the  division,  or  the  quotient,  is  12  with  a  remainder  of  87  ;  or^ 
in  another  form  12x*g^. 

EXAMPLES  EOK  PRACTICE. 

121.  1.     Divide  1121  by  25,  using  as  factors  5  and  5, 

2.  Divide  819  by  -42,  using  as  factors  3,  2,  and  7. 

3.  Divide  1705  by  64,  using  as  factors  8  and  8. 

4.  Divide  4600  by  135,  using  as  factors  3,  5,  3,  and  3. 

5.  Divide  22406  by  125,  using  as  factors  5,  5,  and  5. 

6.  Divide  53479  by  144,  using  as  factors  12  and  12. 

7.  Divide  53479  by  144,  using  as  factors  9  and  16. 
S.  Divide  53479  by  144,  using  as  factors  8  and  18. 
9.  Divide  53479  by  144,  using  as  factors  4,  9,  and  4. 

10.  Divide  53479  by  144,  using  as  factors  4,  3,  3,  and  4. 

11.  Divide  53479  by  144,  using  as  factors  2,  2,  3,  3,  2,  and  2. 

12.  Divide  419047  by  81,  using  as  factors  3,  3,  3,  and  3. 

13.  Divide  341772  by  4095,  using  as  factors  7,  5,  9,  and  13. 
IJf.  Divide  792431  by  72,  using  as  factors  6,  2,  and  6. 

15.     Divide  19111  by  24,  using  as  factors  2,  2,  2,  and  3. 

122.  To  Divide  by  10,  or  any  one  of  its  powers. 

Since  by  the  decimal  system,  numbers  increase  in  value  from  right  to  left  and 
■decrease  from  left  to  right  in  a  tenfold  ratio,  it  follows  tliat  to  cut  off  from  the 
right  of  a  number  one  place,  divides  the  number  by  10,  two  jilaces  by  100,  three 
places  by  luOO,  etc. 

Rule. — From  the  Tight  of  the  dividend  point  off  as  many  orders  of 
units,  or  places,  as  the  divisor  contains  ciphers.  The  figure  or  figures  so 
■cut  off  wiU  express  the  remainder. 

123.  To  Divide  by  any  multiple  of  10,  100,  or  1000,  etc. 
Example.— Divide  16419  by  600. 

FIRST  OPERATION.  EXPLANATION. —  6  and  100  are  factors 

,                  ,  of  600.     First  divide  16419  by  100,  by 

l/OO  )  164/19  separating  from  it  the  last  two  figures, 

obtaining  164  as  the  first  quotient  and  19 

First  quotient  164  — 19,  first  rem.  as  the  first  remainder;   next  divide  164 

by  6  and  obtain  27  as  the  second,  or  last 

SECOND  OPERATION.  quotient,  and  2  as  the  second,  or  last 

6  )  164  remainder  ;   multiply  this  remainder  by 

100,  to  obtain  its  true  value,  and  to  the 

Second  quotient  27. -.2  X  100  =  200,  second  rem.  result  add  the  first  remainderobtaining  219 

rtiA  ^or  the  true  remainder.     The  result  of 

Zly,  true  rem.         ......         .  .-**•«-,         j 

the  division  is  a  quotient  of  27  and  a 

27|^  required  quotient.  remainder  of  219,  or  27Uh 


DIVISION.  33 


B/Ule. — From  the  right  of  the  dividend  separate  as  many  figures  as 
Hhe  divisor  contains  ciphers;  divide  the  figures  at  the  left  of  the  separa- 
trix  by  the  digit  or  digits  of  the  divisor,  and  to  the  remainder,  if  there 
be  one,  annex  the  figures  first  separated  from  the  dividend;  the  result 
will  be  the  true  remainder. 

EXAMPLES   FOR  PRACTICE. 

124-.     1.     Divide  519  by  40,  using  as  factors  4  and  10. 

2.     Divide  1164  by  300,  using  as  factors  3  and  100. 

5.  Divide  2084  by  500,  using  as  factors  5  and  100. 

Jf.  Divide  90406  by  1500,  using  as  factors  15  and  100. 

J.  Divide  83251  by  600,  using  as  factors  6  and  100. 

6.  Divide  416250  by  9000,  using  as  factors  9  and  1000. 

7.  Divide  94275  by  3000,  using  as  factors  3  and  1000. 

8.  Divide  730246  by  11000,  using  as  factors  11  and  1000. 

9.  Divide  50640231  by  120000,  using  as  factors  12  and  10000. 

10.  Divide  620974  by  41000,  using  as  factors  41  and  1000. 

11.  Divide  124689011  by  5910000,  using  as  factors  591  and  10000. 

12.  Divide  365021467  by  6250000,  using  as  factors  625  and  10000. 

MISCELLANEOUS   EXAMPLES   IN   SHORT   DIVISION. 

125.  1.  A  gentleman  left  his  estate  worth  $618330  to  be  shared  equally  by 
his  wife  and  five  children;  what  was  the  sliare  of  each? 

2.  A  -county  containing  400000  acres  is  divided  into  25  townships  of  equal 
area.     How  many  acres  in  each  township? 

3.  $21,735  was  received  from  the  sale  of  a  farm  at  $35  per  acre.  How  many 
acres  did  the  farm  contain? 

Jf.  If  a  speculator  pays  $15730  for  715  acres  of  Nebraska  prairie  land,  and 
sells  the  same  for  $17875,  what  is  his  gain  .per  acre? 

6.  In  New  York  City,  in  February,  1882,  Hazel  walked  660  miles  in  6  days, 
receiving  as  a  prize  $20000.  Allowing  no  time  for  stops,  what  was  his  average 
distance  and  the  average  amount  earned  per  hour? 

6.  Great  Britain  makes  330  million  pins  weekly,  or  9  for  each  inhabitant ; 
what  is  the  number  of  inhabitants? 

7.  The  dividend  is  230304561,  the  divisor  is  15  ;  find  the  quotient  and  the 
remainder? 

8.  The  remainder  is  7,  the  quotient  19023,  and  the  dividend  247306  ;  what  is 
the  divisor? 

9.  If  8  men  can  do  a  certain  piece  of  work  in  9  days,  in  how  many  days  can 
12  men  do  the  same  work? 

10.  I  sell  my  village  home  for  $3250,  my  store  for  $5000,  my  stock  of  goods 
for  $11250,  receiving  in  part  payment  $8775  cash,  and  for  the  remainder  Iowa 
prairie  land  at  $15  per  acre;  ho-w  manv  acres  should  I  receive? 

3 


34  DIVISION. 

11.  The  steamship  Servia  crosses  the  Atlantic  from  New  York  City  to  Liver- 
pool in  150  hours,  averaging  for  the  first  24  hours,  18  miles  per  hour;  for  the 
next  48  hours,  17  miles  per  hour;  for  the  next  30  hours,  19  miles  per  hour;  and 
for  the  next  12  hours,  21  miles  per  hour.  If  the  entire  distance  be  2841  miles, 
what  was  the  average  distance  per  hour  traveled  for  the  remainder  uf  the  time  ? 

Remark. — Short  division,  though  a  mental  process,  is  practicable  whenever  the  divisor  ia 

35  or  less,  if  the  pupil  has  mastered  the  multiplication  table  as  given. 


LONG  DIVISION. 

1*26.  When  the  divisor  is  ;i  number  larger  than  can  be  treated  mentally,  the 
following  method,  called  Long  Division,  is  employed. 

Example.— Divide  81437  by  37. 

Explanation. — Write  the  terms  as  in  short  division,  and  place 
a  line  after  the  dividend  to  separate  it  from  the  quotient,  which  is 
now  to  be  written  at  the  right.  Then  divide  the  first  two  figures 
of  the  dividend,  81,  by  the  divisor,  37,  and  obtain  2  as  the  first 
figure  of  the  quotient ;  then  subtract  from  81  the  product  of  2  x 
37,  or  74,  obtaining  7  as  a  remainder  ;  to  this  remainder  annex  4, 
the  succeeding  figure  of  the  dividend,  which  gives  74  as  the  next 
partial  dividend;  the  divisor  is  contained  in  this  dividend  twice, 
or  2  times,  giving  2  as  the  next  or  second  quotient  figure  ;  sub- 
tracting the  product  of  2  X  37  from  74,  nothing  remains;  then 
bring  down  3,  the  next  figure  of  the  dividend  and  as  it  is  less  than 
the  divisor,  place  a  0  in  the  quotient  ;  next  bring  down  7,  the 
remaining  figure  of  the  dividend  which  gives  37  as  the  last  partial  dividend  ;  the  divisor  is 
contained  in  this  dividend  once,  or  1  time  ;  writing  this  1  as  the  final  figure  of  the  quotient 
and  subtracting  the  last  partial  product  from  the  last  partial  dividend  nothing  remains,  and 
the  quotient,  2201,  is  the  result  of  dividing  81437  by  37. 

Rule. — I.  Write  the  divisor  at  the  left  of  the  dividend  with  a  curved 
line  between  them,  and  another  Hive  at  the  right  of  the  dividend  to  sep- 
arate it  from  the  quotient  when  found. 

II.  From  the  left  of  the  dividend  select  the  least  nuniber  of  figures 
that  will  contain  the  divisor  one  or  more  times,  and  divide.  Write  the 
quotient  figure  thus  obtained  at  the  right  of  the  dividend,  inultiply  the 
divisor  by  this  quotient  figure  and  subtract  the  product  from  the  partial 
dividend  used.  To  the  remainder  annex  the  succeeding  figure  of  the 
dividend  and  divide  as  before;  so  continue  until  the  last  partial  product 
has  been  siobtracted  from  the  last  partial  dividend.  If  there  be  a 
remainder  place  it  over  the  divisor  with  a  line  between,  and  write  the 
resulting  fraction  as  a  part  of  the  quotient. 

Vroof.— Multiply  the  divisor  by  the  quotiext,  and  to  the  product  add  the 
REMAiNDEB  if  there  be  any;  the  result  .should  equal  the  dividend. 


OPERATION. 

Divisor. 

,  Dividend. 

Quotient. 

37 

)  81437 
74 

74 

74 

37 
37 

(  2201 

0 

Remainder. 

127.    Divide 


DIVISION. 
EXAMPLES   IX   LOXG   DIVISION. 


35 


1. 

1728  by  48. 

11. 

115680  by  155. 

21. 

2. 

2025  by  135. 

12. 

29410  by  251. 

22 

3. 

625  by  125. 

13. 

666666  by  2144. 

23. 

U- 

1920  by  160. 

u. 

93462007  by  1525. 

24. 

5. 

;>268  by  45. 

15. 

5005C0500  by  1888. 

25. 

6. 

106295  by  28. 

16. 

21416009  by  5407. 

26. 

7. 

52467  by  109. 

17. 

11460250  by  999. 

27. 

8. 

4762  by  367. 

18. 

87629000  by  11181. 

28. 

9. 

250000  by  793. 

19. 

20405701  by  820006. 

29. 

10. 

87524  by  31. 

20. 

72109904  by  72109. 

30. 

375735212  by  20812. 
26800001  by' 909125. 
104690955  by  5642. 
9000716002  by  1776. 
250252500  by  1562. 
5087910041  by  508791. 
3641694611  by  72853. 
111222333456  by  370054. 
9876543210  by  12345. 
210631890048  by  840263. 


MISCELLANEOUS   EXAMPLES  IN   LONG   DIVISION. 

128.  1.  In  1880  the  total  number  of  persons  engaged  in  all  occupations  in 
the  United  States  was  17392099,  of  which  7670493  were  engaged  in  agriculture; 
how  many  times  greater  is  the  whole  number  of  workers  than  those  engaged 
in  agriculture  ? 

2.  The  2515  miles  of  canal  in  the  United  States  cost  $170028636  ;  what  was 
the  average  cost  per  mile? 

3.  If  an  elephant  produces  120  lb.  of  ivory  and  the  manufactories  of  Sheffield 
consume  yearly  483000  lb.,  how  many  elephants  must  be  killed  each  year  to 
sui)ply  the  Slieffield  market  alone  ? 

4..  In  1880  there  were  in  attendance  in  the  177100  public  schools  of  the  United 
States  9705100  pupils  ;  what  was  the  average  number  in  attendance  in  each 
school  ? 

5.  During  the  financial  crisis  of  1857,  7200  business  houses  in  the  United 
States  failed  for  an  aggregate  of  111  million  dollars  ;  what  was  the  average 
insolvency  ? 

6.  Dan.  Lambert,  at  the  age  of  40,  weighed  739  lb.;  if  his  weight  at  birth  was 
13  lb.,  what  was  his  average  yearly  increase  of  weight  ? 

7.  Between  1871  and  1884  the  Kiml)erly  diamond  field  of  9  acres  produced 
75  million  dollars  wortli  of  diamonds;  what  average  value  per  acre  was  produced 
each  year  ?    Each  month  ? 

8.  A  bottle  thrown  overboard  into  the  Pacific  Ocean  was  picked  uj)  455  days 
later,  6700  miles  distant  from  where  it  was  thrown;  wliat  average  distance  did  it 
float  per  day  ? 

9.  The  great  bell  of  Moscow  weighs  202  tons  of  2240  Ih.  eacli;  if  77  parts  of 
the  metal  of  which  it  is  composed  are  copper  and  the  remaining  23  parts  tin.  liow 
many  pounds  of  each  metal  does  the  bell  contain  ? 

10.  The  log  of  the  yacht  Wanderer  in  circumnavigating  the  globe  in  1880-82, 
showed  48490  miles  run  in  280  days  actual  running  time;  what  was  the  average 
miles  run  per  day  ? 

11.  An  Iowa  firm  manufactures  daily,  from  5  tons  of  paper,  1600  barrels,  of  6  lb. 
weight  each  ;  what  number  of  barrels  can  be  made,  at  this  rate,  from  10750  lb. 
of  paper  ? 


3G  DIVISION. 

t 

12.  On  the  planet  Neptune  60127  days  make  one  year.  A  year  on  Nei)tune 
equals  liow  many  common  years  on  the  earth  ? 

13.  For  the  year  ending  September  30,  1887,  the  exchanges  at  the  Clearing 
House  at  Xew  York  amounted  to  §34872848786,  and  those  of  tlie  36  remaining 
important  cities,  $17253855702.  What  was  the  average  of  the  exchanges  per 
month  at  the  Xew  York  Clearing  House  ?  "Wliat  was  the  average  per  month  of 
the  36  remaining  Clearing  Houses  ? 

IJf.  The  Spanish  Armada,  sent  in  1588,  by  Phillip  II.  of  Spain  for  the  intended 
coufjuest  of  England,  comprised  132  shijis  with  34054  seamen  and  soldiers. 
What  was  the  average  number  with  each  shij)  ? 

15.  In  1885  the  total  loans  of  the  National  Banks  of  Chicago  and  St.  Louis 
were  $55171842,  while  those  of  the  National  Banks  of  New  York  city  were 
$236823598.  How  many  times  greater  was  the  amount  loaned  by  the  banks  of 
New  York  than  by  the  banks  of  the  other  two  cities  named  ? 

16.  The  aggregate  height  above  sea  level  of  the  8  highest  mountains  of  the 
earth,  is  174173  feet.     "What  is  the  average' height  in  miles  of  5280  feet  each  ? 

17.  During  the  year  1854,  50  banks  of  New  York  city  made  exchanges 
through  the  Clearing  House  to  the  amount  of  $5750455987;  and  in  the  year 
1887,  64  banks  nuide  exchanges  to  the  amount  of  $34872848786.  Find  the 
average  clearings  of  each  bank  for  each  of  the  two  years  quoted. 

18.  The  Kingdom  of  Belgium  averages  480  inhabitants  per  square  mile  and 
the  United  States  averages  only  14.  How  many  more  times  densely  peopled  is 
Belgium  than  the  United  States  ? 

19.  The  National  Banks  of  St  Louis  in  1885  made  loans  to  tne  amount  of 
$9182417,  while  those  of  Chicago  made,  during  the  same  year,  loans  to  the 
amount  of  $45089425.  How  many  times  greater  were  the  loans  of  the  banks  of 
Chicago  than  those  of  St.  Louis  ? 

20.  The  total  cost  of  the  railroads  of  the  U.  S.  in  1880  was  $5425772550. 
If  the  average  cost  per  mile  was  $62522,  how  many  miles  had  there  been  built  ? 

21.  In  1880  the  total  railroad  freight  of  the  United  States  was  290897395  tons, 
of  which  42003504  tons  was  grain  and  89622899  tons  was  coal.  How  many  times 
greater  was  the  whole  freight  than  that  of  coal  alone?  How  many  times  greater 
than  that  of  grain  alone  ? 

22.  The  total  expenditures  of  the  railroads  of  the  United  States  in  the  year 
1880,  were  $541950795,  and  their  net  income  was  $119344596.  How  many  times 
greater  were  the  expenditures  than  the  net  income  ? 


AVERAGE.  37 


AVERAGE. 

129.  The  Average  of  several  numerical  terms  is  the  quotient  obtained  by 
dividing   their   sum    by  the   number   of   terms   taken.     Thus,   the  average  of 

32,  40,  56,  IG,  72,  24,  70,  and  66,  is  47,  because  8  times  47  =  376,  which  is  the 
sum  of  the  numbers  taken, 

130.  An  average  may  be  fractional  ;  as  33-|  is  the  average  of  59,  43,  21,  10, 
and  35,  because  the  sum  of  these  five  numbers  equals  5  times  33|. 

Remark. — The  average  numerical  value  of  fractions,  either  common  or  decimal,  may  be 
obtained  by  dividing  the  sum  of  all  such  fractional  expressions  by  the  number  of  such 
expressions  taken. 

Rule. — Divide  the  sum  of  the  terms  &//  the  numher  of  terms  used. 

KXAMFLKS   FOK   PKACTICK. 

131.  Find  the  average  of  the  following  groups  of  numbers  and  prove  the 
results  : 

1.  20,  24,  52,  and  88.  i    3.     71,  46,  200,  11,  93,  51,  and  IT. 

2.  32,  72,  56,  108,  and  144.  |    ^.     5,  28, 19,  72,  40,  85, 106,  29,  and  54. 

5.  A  man  walked  during  six  days  of  a  week,  41,  47,  36,  54,  60,  and  44  miles 
respectively.     How  many  miles  did  he  average  per  day? 

6.  A  merchant  sold  during  the  12  months  of  a  year,  goods  in  amounts  as 
follows:  $14216,  $10008,  $11051,  $11097,  $18241*^  $16900,  813754,  812291, 
$9267,  $12935,  $14901,  and  $20518.     What  were  his  average  sales  per  month? 

7.  An  errand  boy  earned  on  ]\Ionday  73^',  Tuesday  91^-,  Wednesday  49^*,  Thurs- 
day 67^',  Friday  81^',  and  Saturday  95^'.  What  were  his  average  earnings  i)er 
day  for  the  week? 

COMPLEMENT. 

132.  The  Complement  of  a  number  is  the  difference  between  such  number 
and  a  unit  of  the  next  higher  order;  thus,  the  complement  of  6  is  4,  because  4  is 
the  difference  between  6  and  10,  or  1  ten,  a  unit  of  the  next  higher  order  than  G. 

Again,  the  complement  of  83  is  17,  because  17  is  the  difference  between  83 
and  100,  or  1  hundred,  a  unit  of  the  next  higher  order  than  83. 

Again,  the  complement  of  209  is  791,  because  their  sum  is  equal  to  lOdo. 

£XAMPI.£S  FOR   PRACTICE. 

133.  Find  the  complement  of  each  of  the  following  numbers,  and  ])rove  and 
explain  results  : 


1. 

36. 

S. 

115. 

5. 

81. 

1  ''• 

1249. 

y. 

28763. 

2. 

71. 

4. 

704. 

6. 

258. 

1  <^- 

1094. 

10. 

82041. 

38 


PACTOES    AND    FACTORING. 


FACTORS  AND   FACTORING. 

134.  Factors  art-  such  numbers  as  multiplied  together  will  produce  a 
required  number  ;  as  3  and  4.  also  3,  2,  and  2  are  factors  of  12  ;  3  and  15,  also 
5  and  9  are  factors  of  45 

135.  A  Prime  Number  is  one  that  cannot  be  resolved  into  two  or  more 
factors  ;  or,  it  is  a  number  exactly  divisible  only  by  itself  and  unity;  thus,  2,  3, 
5,  7,  11,  and  13,  are  prime  numbers.     2  is  the  only  even  number  that  is  prime. 

13C.     A  Composite  Number  is  one  that  can  be  resolved  into  factors. 

137.     A  Prime  Factor  is  n  jirime^nwmhev  used  as  vi  factor. 
To  aid  the  pujjil  in  determining  the  prime  factors  of  a  composite  number  we 
give  the  following 

Table  of  Prime  Numbers  from  1  to  lOOO. 


1 

59 

139 

233 

337 

439 

557 

653 

769 

883 

2 

r,i 

149 

239 

347 

443 

563 

659 

773 

887 

3 

07 

151 

241 

349 

449 

569 

661 

787 

907 

5 

71 

157 

251 

353 

457 

571 

673 

797 

911 

7 

73 

163 

257 

359 

461 

577 

677 

809 

919 

11 

79 

1G7 

263 

367 

463 

587 

683 

811 

929 

13 

83 

173 

269 

373 

467 

593 

691 

821 

937 

17 

89 

179 

271 

379 

479 

599 

701 

823 

941 

19 

97 

181 

277 

383 

487 

601 

709 

827 

947 

23 

101 

191 

281 

389 

491 

607 

719 

829 

953 

2!) 

103 

193 

283 

397 

499 

613 

727 

839 

967 

31 

107 

197 

293 

401 

503 

617 

733 

853 

971 

37 

109 

199 

307 

409 

509 

619 

739 

857 

977 

41 

113 

211 

311 

419 

521 

631 

743 

859 

983 

43 

127 

223 

313 

421 

523 

641 

751 

863 

991 

47 

131 

227 

317 

431 

541 

643 

757 

877 

997 

53 

137 

229 

331 

433 

547 

647 

761 

881 

Remauk.-  The  pupil  can  with  little  labor  memorize  the  prime  numbers  from  1  to  100. 


It^ 


FACTORS   AND    FACTORING.  39 

138.     To  Find  the  Prime  Factors  of  a  Composite  Number. 
Example. — Find  the  prime  factors  of  4290. 

■OPERATION. 

5  )  4290         Explanation.— Observe  that  the  given  number  ends  with  a  cipher,  hence  is 
— :; — r~      exactly  divisible  by  the  prime  number  5,  by  which  divide  it;  next,  observe  that 

2  J^oo      ^jjg  quotient  ends  with  an  even  number,  and  is,  therefore,  exactly  divisible  by 

3  )  429      2,  so  divide  by  2  ;  then  observe  that  3  will  exactly  divide  the  quotient  429 ; 
-|^  -J  143      divide  by  it,  obtaining  143,  which  divide  by  11,  obtaining  13,  which  divided  by 

— '-     itself,  gives  a  quotient  of  1.    All  the  divisors  being  prime  numbers  they  together 

13  )  13      constitute  the  prime  factors  of  4290. 
1 

Rule. — Divide  by  any  prime  niomber  that  is  exactly  contained  in  the 
dividend;  divide  the  resulting  qiooiient  in  the  same  manner,  and  con- 
tinue this  until  the  final  quotient  is  1-  The  prime  divisors  will  he  aU 
the  prime  factors  of  the  dividend. 


KXAMPLKS   FOK  PRACTICE 

139.    Resolve 

1.  27  into  its  prime  factors. 

2.  117  into  its  prime  factors. 
S.     165  into  its  prime  factors. 
Jf.     93  into  its  prime  factors. 
J.     2376  into  its  prime  factors. 


6'.  1050  into  its  prime  factors. 

7.  144  into  its  prime  factors. 

S.  15625  into  its  prime  factors. 

9.  22464  into  its  prime  factors. 

10.  881790  into  its  prime  factors. 


DIVISORS. 

140.  An  Exact  Diyisor  of  a  number  is  one  which  will  divide  it  without  a 
remainder,  or  which  gives  a  whole  number  as  a  quotient ;  thus,  5  is  an  exact 
divisor  of  15,  3  of  12,  and  2  of  4. 

141.  1.  Any  number  is  divisible  by  itself  and  1. 
3.  Any  even  number  is  divisible  by  2. 

3.  Any  number  ending  with  5  or  0  is  divisible  by  5. 

4.  Any  number  ending  with  0  is  divisible  by  10. 

5.  An  even  number  is  not  an  exact  divisor  of  an  odd  number. 

6.  A  composite  number  is  an  exact  divisor  of  any  number  Avhen  all  its  factors 
are  divisors  of  the  same  number. 

142.  A  Common  Divisor  of  two  or  more  numbers  is  one  that  will  exactly 
divide  all  the  numbers  considered;  thus  3  is  a  common  divisor  of  6,  9, 12,  and  15; 
also  7  is  a  common  divisor  of  14,  28,  35,  and  49. 

143.  The  Greatest  Common  Divisor  of  two  or  more  numbers  is  the 
greatest  number  that  is  exactly  contained  in  all  of  them,  or  that  will  divide  each 
of  them  without  a  remainder. 

144.  Numbers  having  no  common  divisor,  or  factor,  are  said  to  be  prime  to 
€ach  other. 


40  DIVISORS. 

145.     To  Find  the  Greatest  Common  Divisor. 

I.  When  the  numbers  are  readily  factored. 

Example. — Find  the  greatest  common  divisor  of  10,  15,  and  35. 

OPERATION.  Explanation. — By  inspection  find  that  the  prime  number  5  is  ai» 

r  \  -iQ -IS o-      exact  divisor  of  each  of  the  numbers  given;  using  it  as  a  divisor, 

gives  as  quotients  2,  3,  and  7;  these  being  prime  numbers  have  no- 

2  —    3 —    7      common  divisor,  therefore  5  is  a  common  divisor  of  the  numbers  10, 
15,  and  35.  and  as  it  is  the  greatest  number  that  will  exactly  divide 
them  it  must  be  their  greatest  common  divisor. 

Remark.  —  When  it  is  determined  by  inspection  that  any  composite  number  will  exactly 
divide  all  the  numbers  of  which  we  wish  to  obtain  the  greatest  common  divisor,  such  com- 
posite number  may  wisely  be  used  as  a  divisor. 

II.  When  numbers  are  less  readily  factored. 

Example. — Find  the  greatest  common  divisor  of  140,  210,  350,  -420,  and  630. 

OPERATION.  Explanation.  —  To  prevent  confusion,  sepa. 

^>\■.^/^       «,^       o-/^        . -./^       ,..-./i        rate  the  numbers  by  a  short  dash.     Observe  that 
2)140  —  210  —  3o0  —  420  —  630       „     .„  .,     ■..  .i         u    c  ,u  k       n 

J_ 2  will  exactly  divide  each  of  the  numbers,  like- 

g  \  f-Q 2Q^ -irrx .-)]() 315        wise  that  5  and  7  will  exactly  divide  the  successive 

quotients ;  therefore  divide  by  2,  5,  and  7  ;  then 

T  )  14 —    21 —   35 —   42  — 105        observethattheremainingquotients,  2, 3, 5, 6,  and 

-— 15  have  no  common  divisor  ;  hence  the  divisors 

''^  —      3  —      .T  —      6  —    15        2,  5,  and  7  are  all  factors  of  the  greatest  common 

divisor,  which  is  70. 

Rule. — I.  Write  the  nuvibers  in  a  horizontaZ  line,  separating  thein  hij 
a  dash. 

n.  Divide  by  any  number  that  mill  exactly  divide  all  the  numbers 
given,  and  so  continue  until  the  quotients  have  no  common  divisor. 

HI.    Multiply  together  the  divisors  for  the  Greatest  Common  Divisor- 

Remark. — When  factors  cannot  be  readily  determined  by  inspection  the  numbers  may  be 
resolved  into  their  prime  factors.  The  product  of  all  the  common  factors  of  all  the  givea 
numbers  will  be  the  greatest  common  divisor. 


EXAMPLKS  FOR   PRACTICE. 

146.     Find  the  greatest  common  divisor  of 


1.  22,  55,  and  99. 

2.  24,  36,  60,  and  96. 

3.  32,  48,  80,  112,  and  144. 

A.  54,  72,  90,  126,  180,  and  216. 


7.  252,  630,  1134,  and  1456. 

8.  2150,  600,  3650,  1000,  and  oOO. 

9.  302,  453,  755,  1057,  and  1661. 
10.  126,  441,  567,  693,  and  1071. 


5.  104,  156,  260,  364,  and  572.  11.     210,  350,  280,  840,  and  1260. 

6.  135,  450,  315,  and  585.  I  12,     200,325,  525,  350,  and  675. 


MULTIPLES. 


41 


147.     "When  no  Common  Factor  can  be  Determined  by  Inspection 

Example. — What  is  the  greatest  common  divisor  of  182  and  858. 

OPERATION. 


182 
130 

52 
52 

0 


858 
728 

130 
104 

26 


Explanation. — Draw  two  vertical  lines  and  write  the  numbers  on 
the  right  and  left.  Then  divide  858  by  182,  and  write  the  quotient, 
4,  between  the  lines;  then  divide  182  by  the  remainder,  130,  and  write 
the  quotient,  1,  between  the  lines;  next  divide  130  by  52  and  write  the 
quotient,  2,  as  before;  next  divide  52  by  26  and  write  the  quotient  as 
before.  As  there  is  nothing  now  remaining  the  last  divisor,  26,  is  the 
greatest  common  divisor  of  the  given  numbers. 


Remarks. — 1.  The  greatest  common  divisor  of  several  numbers  which  cannot  be  factored, 
may  be  obtained  by  taking  any  two  of  them  and  applying  the  above  formula;  then  the  divisor 
thus  obtained  and  one  of  the  remaining  numbers,  and  so  on  until  the  last.  If  1  be  the  final 
result  they  have  no  common  divisor;  if  any  number  greater  than  1,  that  number  must  be  the 
greatest  common  divisor  of  all  the  given  numbers. 

2.  The  only  practical  use  of  the  Greatest  Common  Divisor  is  in  the  reduction  of  a  common 
fraction  to  its  lowest  terms;  we  thus  find  a  number  that  will  affect  such  reduction  by  a  division 
of  the  terms  but  once. 

Rule. — Divide  the  greater  numher  by  the  less,  the  divisor  by  the 
remainder,  and  to  continue  until  nothing  remains.  The  last  divisor  will 
be  the  Greatest  Common  Divisor. 


148. 

1.  316  and  664. 

2.  96  and  216. 

3.  1226  and  2722 

4.  1649  and  5423 


KXAMPLES  roil   PKACTICK 

Find  the  greatest  common  divisor  of 
J.     1377  and  1581. 

6.  92  and  124. 

7.  679  and  1869. 
<^.  2047  and  3013. 


9. 

231  and  273. 

10. 

1179  and  1703 

11. 

1888  and  1425 

12. 

1900  and  1375 

MULTIPLES. 

149.  A  Multiple  is  a  number  exactly  divisible  by  a  given  number;  as,  12  is 
a  multiple  of  6. 

150.  A  Commou  Multiple  is  a  number  exactly  divisible  ])y  two  or  nioie 
given  numbers;  as,  12  is  a  common  multiple  of  6,  3,  and  2. 

151.  The  Least  Commou  Multiple  of  two  or  more  numbers  is  the  least 
number  exactly  divisible  by  each  of  them;  as,  36  is  the  least  common  multiple 
of  18,  9,  6,  4,  3,  and  12.  ^ 

152.  Principles. — l.  The  2^roduct  of  two  or  more  numbers,  or  any  number 
of  tim"s  their  product,  must  be  a  common  multiple  of  the  numbers. 

2.  Two  or  more  numbers  may  have  any  number  of  common  multiples. 

3.  A  multijjle  of  a  number  must  contain  all  the  prime  factors  of  that  number. 
4-   T/ie  common  midtiple  of  several  numbers  must   contain  all  the  factors  of 

all  the  numbers. 

5.  The  least  common  multiple  of  ttvo  or  more  numbers  is  the  least  number 
that  will  contain  all  the  prime  factors  of  the  numbers  yiven. 


42  MULTIPLES. 

153.     To  Find  the  Least  Common  Multiple  of  Two  or  More  Numbers 
Example. — Find  the  least  common  multiple  of  12,  16,  63,  and  90. 

Explanation. — By  factoring,  find  the  prime  factors  of  12  which  are  2,  2,  and  3. 

"    16  "       2,  2,  2,  and  2. 

"   63  "       3,  3,  and  7. 

"  90  "       3,  3,  2,  and  5. 

Since  no  number  less  than  90  can  be  divided  by  90,  it  is  evident  that  the  least  common  multiple 
cannot  be  less  than  that  number  ;  hence  it  must  contain  3,  3,  2,  and  5,  the  factors  of  90 ; 
including  with  these  another  2,  gives  all  the  factors  of  12;  two  more  2's  all  the  factors  of  16  ; 
and  if  7  be  included,  all  the  factors  of  63  are  obtained  ;  hence  the  product  of  the  factors  3,  3, 
2.  5,  2,  2,  2,  2,  and  7,  or  5040  must  be  the  least  common  multiple  of  the  numbers  12,  16,  63, 
and  90. 

The  method  of  determining  tlie  least  common  multiple  by  formula  given  below, 
will  be  found  convenient. 

Example. — Find  the  least  common  multiple  of  the  numbers  12, 16,  63,  and  90. 

Write  the  numbers  in  a  horizontal  line  to  obviate  confusion,  and  separate  them 
by  a  dash. 

OPERATION.  Explanation.— First  divide  by  2  ;  63  not  being  divisible  by  2 

2  \  12 16 63 90  bring  it  to  the  lower  line  and  divide  again  by  2;  neither  63  nor  45 

being  divisible  by  2,  bring  both  to  the  lower,  or  quotient  line. 

2  )  6 —  8 — 63 — 45  Next  divide  by  3;  4  not  being  divisible  by  3,  bring  it  to  the  quo- 
q  \  Q        4     63      15  *^®°^  ^'°*^  ^°*^  divide  again  by  3;  the  remaining  numbers  4,  7,  and 

5  being  prime  to  each  other,  are  to  be  taken,  together  with  the 

3  )  1 —  4 — 21 — 15       prime  divisors  2,  2,  3,  and  3,  as  factors  of  the  least  common  mul- 

~        T       I       I      tiple;  their  product  is  5040,  the  same  as  before  found. 

Remarks — 1.  This  principle  has  a  practical  value  only  in  determining  the  least  common 
denominator  of  common  fractions,  and  is  even  then  rarely  used. 

2.  Where  one  of  the  numbers  given  is  a  factor  of  another,  reject  the  smaller. 

3.  When  it  is  observed  that  any  composite  number  is  exactly  contained  in  all  the  numbers 
given,  divide  by  such  composite  number  rather  than  by  its  prime  factors;  the  operation  will 
thus  be  shortened. 

Rule. — 1  Write  the  nunibers  in  a  horizontal  line,  separating  them  by 
a  dash. 

11.  Divide  by  any  factor  common  to  all  the  numbers,  or  by  any  prime 
factor  of  any  two  or  more  of  them.  In  the  same  manner  divide  the 
quotients  obtained,  and  continue  until  the  quotients  are  prime  to  each 
other. 

in.  The  product  of  the  divisors  and  prime  remainders  is  the  Least 
Common  Multiple. 

154.     Greatest  Common  Divisor  and  Least  Common  Multiple  Compared. 

I.  Tlie  greatest  common  divisor  is  the  product  of  all  the  pj^me  factors  common 
to  all  tlie  numbers. 

II.  The  least  common  multiple  is  the  product  of  all  the  prime  factors  of  all 
the  numbers. 


CANCELLATION.  43 


EXAMPLES  FOR   PRACTICE 

155.     Find  the  least  common  multiple  of 
1.     12,  20,  and  32. 


2.  25,  90,  and  225. 

3.  6,  16,  26,  and  36. 


4.  42,  210,  56,  and  35.    I    7.  18,  80,  99,  and  120. 

5.  5,  30,  24,  and  28.        I   8.  2,  3,  4,  5,  6,  7,  and  8. 

6.  11,  32,  216,  and  66.    I    9.  21,  72,  24,  and  30. 

CANCELLATION. 


156.  Cancellation  is  the  omission  of  the  same  factor  from  terms  sustaining 
to  each  other  the  relation  of  dividend  and  divisor.  It  is  used  for  the  purpose  of 
saving  labor  in  division,  and  is  an  application  of  the  principle  already  given, 
that  dividing  both  dividend  and  divisor  by  the  same  number  will  not  alter  the 
quotient;  thus  f  may  be  read  2  divided  4;  divide  both  terms  by  2  and  the  result 
is  1  divided  by  2,  or  -|. 

2  X  27 
Again,  mav  be  read  2  times  27,  divided  bv  4  times  18  ;  reiecting  the 

4  X  18        " 
factor  2  from  the  2  in  the  dividend  and  from  the  4  of  the  divisor,  also  the  factor 

9  from  the  27  of  the  dividend  and  the  18  of  the  divisor,  gives =■ : — , 

Mxi^2       2x2 

or  f,  or  3  divided  by  4,  as  a  final  quotient. 

The  correctness  of  this  result  is  easily  proved  by  factoring  the  dividend  and 

divisor,  thus  : = ,  then  reiecting  2  and  9  from  both  terms, 

4  X  18       2  X  2  x  9  X  2 

or  cancelling,  obtain =  f  Ans. 

^  X  2  X  0  X  2 

157.  We  may  supplement  the  former  definition  thus:  The  rejection  of  equiva- 
lents of  factors  from  terms  sustaining  to  each  other  the  relation  of  dividend  and 
divisor,  is  cancellation. 

Example.  —What  is  the  quotient  of  3x2x28x5x7x51  divided  bv 
6  X  11  X  4  X  7  X  35  X  17? 

OPERATION.  Explanation.— Cancel  6  from  the  divisor  and 

$x2x2$X^XlxW3  3x2  from  tlie  dividend;  4x7  from  tlie  divisor 

'■ =  y5j.        and  28  from  the  dividend;  the  35  from  the  divisor 

^  XllX'ixIl  XUXU  and  5x7  from  the  dividend;  the  17  from    the 

divisor  and  the  51  from  the  dividend,  leaving  3  in 
the  dividend,  and  11  in  the  divisor;  the  quotient  is  j\. 

Remark.  — This  principle  can  be  put  to  frequent  and  valuable  use  in  a  great  variety  of 
business  computations. 

Rule. — I.  Write  the  divisor  helow  the  dividend  iinth  a  line  separating 
them. 

II.  Cancel  from  the  dividend  and  divisor  all  factors  common  tu  both; 
then  divide  the  product  of  the  remaining  factors  of  the  dividend  hy  the 
product  of  the  remaining  factors  of  the  divisor. 


44  CANCELLATION. 

EXAMPLES  FOK   PRACTICE. 

158.     1.  Determine  by  cancellation  the  quotient  of  5  x  9  x  2  x  13  x  40  x  6 
divided  bv  8  x  3  x  7  X  26. 

2.  Determine  by  cancellation  the  quotient  of  64  x  25  x  3  x  15  divided  by 

45  X  12  X  4  X  11  X  36. 
In  like  manner, 

3.  Divide  210x  9x  78x  5  x23  X  10  X  36  by  13x144x40x3x27 X5x400. 

4.  Divide  38  X  4  X  55  X  9  x  32  X  30  by  12  X  11  X  3  X  16  x  19  x  5. 

5.  Divide  51  X  7  X  9  X  27  X  40  X  54  by  63  X  17  X  9  x  200. 

6.  Divide  24  X  25  X  26  X  27  by  2  x  4  x  5  x  9  x  13. 

7.  Divide  2  X  3  X  4  x  5  x  6  x  7  X  8  x  9  by  23  x  45  x  67  x  89. 

8.  Divide  the  product  of  the  numbers  98,  76,  54,  and  32  l)y  the  product  of 
the  numbers  9,  8,  7,  6,  5,  4,  3,  and  2. 

9.  Divide  the  product  of  33,  4,  42,  9,  5,  and  60  by  the  product  of  7,  15,  12, 
and  11. 

10.  Divide  the  product  of  416,216,  and  810  by  the  product  of  135,  52,  24, 
and  5. 

11.  How  many  bushels  of  potatoes  at  60^  per  bushel  will  pay  for  450  lb.  of 
sugar  at  6^  per  pound  ? 

12.  A  farmer  traded  4  hogs  weighing  325  lb.  each,  at  6^  })er  pound,  for  sugar 
at  5^  per  pound.  How  many  entire  barrels  of  312  lb.  each  should  the  farmer 
receive  ? 

IS.  I  bought  18  car  loads  of  apples  of  216  barrels  each,  each  barrel  containing 
3  bushels  at  60^  per  bushel,  and  paid  for  the  same  in  woolen  cloth.  If  each 
bale  of  cloth  contained  600  yd.  at  30  cents  per  yard,  how  many  bales  and  how 
many  odd  yards  did  I  deliver  ? 

IJ^  How  many  yards  of  cloth  at  .15^  per  yard  should  be  given  for  9  barrels  of 
pork  of  200  lb.  each,  at  6'/  per  pound  ? 

15.  A  hunter  traded  6  dozen  coon-skins  at  40^  each,  for  powder  at  75^  per  lb. 
How  many  5  lb.  cans  of  powder  should  he  receive  ? 

16.  How  many  pieces  of  cloth  of  45  yd.  each,  should  be  received  for  5  baskets 
of  eggs,  each  basket  containing  21  dozens  at  18^'  per  dozen,  if  the  cloth  be  valued 
at  8^  per  yard  ? 

17.  How  many  quarter  sections  of  Kansas  prairie  land  valued  at  §9  per  acre, 
should  be  received  for  80  cattle  worth  878  per  head  ? 

Rem.vbk. — A  section  of  land,  in  the  Unitefl  States,  contains  640  acres. 

18.  How  many  years'  work  of  12  months  of  26  days  each,  must  be  given  for 
a  farm  of  112  acres  at  |i78  per  acre,  if  labor  be  worth  $2  yer  day.'' 

19.  A  farmer  exchanged  3  loads  of  oats,  each  load  containing  27  sacks  of  2 
bushels  each,  worth  33^  per  bushel,  for  flour  at  6'/  per  pound.  At  196  lb.  per 
barrel,  how  many  barrels  should  he  have  received  ? 

20.  How  many  sections  of  Texas  jirairie  land  at  $8  per  acre  should  be  given 
for  ap  Ohio  farm  of  272  acres  at  $45  per  acre  .'' 


FEACTIONS.  45 


FRACTIONS. 

159.  A  Fraction  is  one  or  more  of  the  equiil  parts  of  a  unit.  If  a  unit  be 
divided  into  3  equal  parts,  one  of  the  parts  is  called  one-third  and  is  written  ^  ; 
two  of  the  parts  are  called  two-thirds  and  are  written  §. 

160.  A  Fractional  Unit  is  one  of  the  equal  parts  into  which  the  number  or 
thing  is  divided.     \,  \,  ^,  are  fractional  units. 

161.  The  Numerator  is  the  number  above  the  line;  it  numerates,  or  num- 
bers the  parts,  and  is  a  dividend, 

162.  The  Denominator  is  the  number  below  the  line;  it  denominates,  or 
names  the  value,  or  size,  of  the  parts  showing  the  number  of  parts  into  which 
the  unit  has  been  divided.     It  is  a  divisor. 

163.  The  Terms  of  a  fraction  are  the  numerator  and  denominator,  taken 
together. 

164.  The  Value  of  a  fraction  is  the  quotient  of  the  numerator  divided  by 
the  denominator. 

165.  Fractions  are  distinguished  as  Common  Fractions  and  Decimal  Frac- 
tions; and  common  fractions  are  either  jorojoer  or  Improper. 

166.  A  Common  Fraction  is  one  expressed  by  two  numbers,  one  written 
above  the  other,  Avitli  a  line  between. 

167.  A  Proper  Fraction  is  one  whose  value  is  less  than  1,  the  numerator 
being  less  than  the  denomimdor.     \,  f,  \,  f,  ^i,  -^  ^vo. p)roper  fractions. 

168.  An  Improper  Fraction  is  one  whose  numerator  is  either  equal  to  or 
greater  than  its  denominator;  its  value  is  equal  to  or  greater  than  1.  |,  |,  \,  -|, 
V'  St  '  rt  ^^"6  improper  fractions. 

169.  A  Mixed  Number  is  an  entire  or  luhole  number  and  n  fraction  united. 
2i,  5f,  91,  14|3-,  lOTfl  are  mixed  numbers. 

170.  A  Complex  Fraction  is  one  having  a  fraction  for  its  numerator  or 

denominator,  or  for  both  of  its  terms. 

As  a  fraction  indicates  a  division  to  be  performed,  a  complex  fraction  indicates  a  division  of 
fractions  to  be  performed.     ^  ^^  *  complex  fraction  and  indicates  that  f  is  to  be  divided  by  S  ; 

'    5  * 

the  expression  is  read  f  s-  5  ;  — ,  and  ^  are  also  complex  fractions. 

i  8 

Principles. — 1.  Multiplying  the  numerator  multiplies  the  fraction;  dividing 
the  numerator  divides  the  fraction. 

2.  Multiplyhuj  the  denominator  divides  the  fraction;  di aiding  the  denominator 
multiplies  the  fraction. 

3.  Multiplying  or  dividing  both  terms  of  a  Jradioti  Inj  the  same  number  does 
not  change  the  value  of  the  fraction. 


46  REDUCTION    OF   FRACTIONS. 

REDUCTION   OF    FRACTIONS. 

171.     To  Reduce  a  Whole  Number  to  a  Fractional  Form. 

Example. — Reduce  3  to  ;i  fraction  the  denominator  of  wliicli  is  7. 

Explanation.— The  fractional  unit  having  7  for  a  denominator  is  4  ;  and  since  1  unit 
equals  7  sevenths,  3  units  which  are  3  times  1  unit  must  equal  3  times  7  sevenths,  or  21  sev- 
enths ;  therefore,  3  =  V- 

Rule. — Multiply  the  ichole  number  hy  the  required  deiwrninator,  and 
place  the  product  over  the  denominator  for  a  numerator. 

EXAMPLES  FOR  PRACTICE. 

Wi.     1.  Reduce  5  to  a  fraction  the"  denominator  of  which  will  be  4. 

J.  Reduce  7  to  a  fraction  the  denominator  of  which  will  be  9. 

J.  Reduce  4  to  a  fraction  the  denominator  of  Avhich  will  be  13. 

,'/.  Reduce  3  to  a  fraction  the  denominator  of  which  will  l)c  8. 

').  Reduce  8  to  a  fraction  the  denominator  of  Avhich  will  bo  12. 

6.  Reduce  15  to  a  fraction  the  denominator  of  which  will  bo  10. 

7.  Reduce  14  to  a  fraction  the  denominator  of  which  will  be  5. 

8.  Reduce  27  to  a  fraction  the  denominator  of  which  will  1)e  11. 

9.  Reduce  416  to  a  fraction  the  denominator  of  which  will  be  23. 
JO.  Reduce  1125  to  a  fraction  the  denominator  of  which  Avill  be  57. 

173.  To  Reduce  a  Mixed  Number  to  an  Improper  Fraction. 
Example. — Reduce  5f  to  an  improper  fraction. 

Explanation.— Since  1  unit  is  equal  to  3  thirds,  5  units,  whicli  are  5  times  1  unit,  must 
be  equal  to  5  times  3  thirds,  or  15  thirds;  and  15  thirds  plus  2  thirds  equals  17  thirds;  there- 
fore, ^  =  y. 

Rule. — Multiply  the  whole  number  by  the  denominator  of  the  fraction, 
to  the  product  add  the  numerator,  and  place  the  sum  over  the  denom- 
inator. 

KXAMPLES    POK    I'KAi'TICK. 

174.  Reduce 


1.  3^  to  an  improper  fraction. 

2.  7|  to  an  impro])er  fraction. 

3.  10|  to  an  improper  fraction. 
Jf..  43f  to  an  improjier  fraction. 
J.  16^  to  an  improjyer  fraction. 


'j.  78f  to  an  improper  fraction. 

7.  ^^^i^  to  an  improper  fraction. 

8.  170^L.  to  an  improper  fraction. 
•9.  KMOg'y  to  an  im])roper  fraction. 

]i>.  96Sj^  1o  an  improper  fraction. 


175.     To  Reduce  an  Improper  Fraction  to  a  Whole  or  Mixed  Number. 

Ex.\MPLE.  —  Reduce  -/  tu  a  whole  or  mixed  number. 

Explanation. — Since  4  fourths  make  1  unit,  23  fourtlis  will  make  as  mrny  vmits  a.s  4  is 
contained  times  in  23,  or  5  times  with  a  remainder  of  3,  or  three-fourths;  therefore,-,"  =  5f. 


REDUCTION    OF    FRACTIONS.  47 

Rule. — Divide  the  numerator  by  the  dertominator,  place  the  remain- 
der, if  any,  over  the  denominator,  and  annex  the  fraction  thus  found 
to  the  entire  part  of  the  quotient 


KXAMPtES  FOK  PRACTICE 

176.     Reduce 

1.  1^  to  a  whole  or  mixed  number. 

2.  Y  to  a  whole  or  mixed  number. 

3.  ^  to  a  whole  or  mixed  number. 
4..     ^^  to  a  whole  or  mixed  number. 


5.     y^^  to  a  whole  or  mixed  number. 


6.  \y>  to  a  whole  or  mixed  number. 

7.  i||i  to  a  whcle  or  mixed  number. 

8.  |-|  to  a  Avhole  or  mixed  number, 
^-  Vs¥  ^^  ^  whole  cr  mixed  number. 

10.  1  Iff"  to  a  whole  or  mi  xed  number. 


177.     To  Reduce  a  Fraction  to  its  Lowest  Terms. 
Example. — Reduce  y\  to  its  lowest  terms. 

Explanation. — By  applying  the  principles  of  factoring,  change  the  form  of  the  fraction 

2x3 
JL  to  — ~ ;  then  by  cancellation  reject  the  2  and  3  from  the  numerator,  and  the  same 

2X3X3 
factors  from  the  denominator,  leaving  1  for  the  new  numerator  and  3  for  tlie  new  denom- 
inator;  the  resulting  fraction  is  i. 

Or,  observe  that  6  is  a  factor  of  both  the  terms  and  that  i  is  the  result  of  dividing  both 
the  terms  by  6. 

Rules. — 1.  Divide  both  terms  of  the  fraction  hy  their  greatest  common 
■divisor.    Or, 

2.  Divide  both  terms  of  the  fraction  hy  any  common  factor,  and  con- 
tinue the  operation  ivith  the  resulting  fractions  until  they  have  no  com- 
mon divisor. 

Remarks. — 1.  When  the  terms  of  a  fraction  have  no  common  factor,  the  fraction  is  in  its 
simplest  form,  or  its  lowest  terras. 

2.  If  both  terms  of  a  fraction  be  divided  by  their  greatest  common  divisor  the  fraction  will 
be  reduced  to  its  loirest  terms.  This  is  the  only  use  in  practical  arithmetic  of  the  theory  of 
the  greatest  common  divisor. 

EXAMPLES  FOK   PRACTICE. 


178.  1.  Reduce  |^  to  its  lowest  terms. 

2.  Reduce  ^f  to  its  lowest  terms. 

3.  Reduce  |-§-  to  its  bwest  terms. 
Jf.  Reduce  -J-f  to  its  lowest  terms. 
0.  Reduce  -^-^  to  its  lowest  terms. 


6'.  Reduce  -^^^  to  its  lowest  terms. 

7.  Reduce  -^^  to  its  lowest  terms. 

8.  Reduce  ||||  to  its  lowest  terms. 

9.  Reduce  ||^  to  its  lowest  terms. 
10.  Reduce  m  to  its  lowest  terms. 


179.     To  Reduce  a  Fraction  to  Higher  Terms. 

Example. — Reduce  f  to  a  fraction  the  denominator  of  which  is  21. 

Explanation. — Since  7  is  contained  in  21  three  times,  the  given  fraction  maybe  reduced  to 
a  fraction  whose  denominator  is  21,  by  multiplying  both  of  its  terms  by  3;  multiplying  22L? 
gives  l\,  the  required  result.     This  operation  does  not  alter  the  value  of  the  given  fraction. 

Rule. — Divide  the  required  denominator  hy  the  denominator  of  the 
given  fraction  and  multiply  the  niimcrator  hy  the  quotient  thus  obtained; 
write  the  product  over  the  required  denominator. 


48  REDl'CTIOX    OF    FRACTIONS. 

EXAMPLES   FOR   PRACTICE. 

180.  1.  Reduce  f  to  a  fraction  the  denominator  of  whicli  is  15. 
2.  Reduce  ^  to  a  fraction  the  denominator  of  which  is  36. 
S.  Reduce  -^  to  a  fraction  the  denominator  of  which  is  42. 
J^.  Reduce  f  to  a  fraction  the  denominator  of  which  is  32. 

5.  Reduce  -^j  to  a  fraction  the  denominator  of  which  is  88. 

6.  Reduce  -^  to  a  fraction  the  denominator  of  whicli  is  52. 

7.  Reduce  4^  to  a  fraction  the  denominator  of  which  is  115. 
S.  Reduce  -^  to  a  fraction  the  denominator  of  which  is  128. 
0.  Reduce  /|-  to  a  fraction  the  denominator  of  which  is  102. 

10,      Reduce  l\  to  a  fraction  the  denominator  of  which  is  147. 

181.  To  Reduce  Fractions  to  Equivalent  Fractions  Having  a  Common  Denom- 
inator. 

Example. — Reduce  f,  ^,  -|,  4,  to  equivalent  fractions  having  a  common 
denominator. 

EXPLAXA.TION. — The  product  of  the  denominators  3,  2,  5,  7,  =  210,  and  this  number  is 
exactly  divisible  by  each  of  the  several  denominators;  hence  each  of  the  given  fractions  may 
be  reduced  to  an  equivalent  having  210  for  a  denominator;  the  desired  result  is  then  accom- 
plished, as  210  is  a  denominator  common  to  all  the  given  fractions;  f  =  ^*g,  |  =  iyg,  |  =  \\^, 

«nH    4  —  ISO 

ana  ,  —  jto- 

Rule. — Multiply  together  the  deivominators  of  the  given  fractions  for 
a  common  deiiominator.  Multiply  each  jiuDveT'ator  hy  all  the  denomina- 
tors except  its  aim  and  irrife  the  several  results  iti  turn  over  the  common 
denominator. 

Remark. — "Where  one  or  more  of  the  given  denominators  are  factors  of  the  others,  the 
smaller  may  he  rejected. 

EXAMPLES  FOR   PRACTICE. 

182.  Reduce  to  e<iuivalent  fractions  having  a  common  denominator: 


1'  h\,\,  and  -|. 

^-  -J,  iV.  f  4>  i.  and -iV 

3.  A»  h  \,  2,  tV.  h  and  -|. 

-4.  n,  5,  h  \h  h  ■i\,  and  -J. 

^.  fa,  51,  t3„,  38i,  23|,  and]2. 


6.  44,  5i,  13if,  0,  and  11  J. 

' •  h  I,  II,  \h  If,  and  \. 

8.  2^,  7-2,llt,  23^V,  i  I,  and5. 

9'  tl,  f,  fj,  8,  iiil^,  andf 

10.  A,  4,  ^l,  -j^^gV,  H,  t  and  20. 


1S:J.  To  Reduce  Fractions  to  Equivalent  Fractions  Having  the  Least  Com- 
mon Benominator. 

The  Least  Common  Denominator  of  two  or  more  fractions  is  the  least  denom- 
inator to  which  tlu'V  can  all  Ije  reduced,  and  must  be  the  least  common  multiple 
of  the  given  denominators. 

Example. — Reduce  \,  |,  I,  \,  -^^,  and  ^  to  equivalent  fractions  having  the 
least  common  denominator. 

OPERATION.  Explanation. —Find  the  least 

3,  2,  )  9  —  15  —  4                   \  =  -j3g^.      ^  =  -jSJi^.  common  multiple  of  the  given  de- 

3         5        2                   i  ^=  J4-''       4  =    "-"  nominators  for  the  least  common 

igo-      L  —  isu-  denominator,  which  is  180.     Then 

3X2X3X5X2  =  180.     -j^  =  ,%■      f  =  i-J;;.  by  Art.  179,  reduce  each  of  the 

^ven  fractions  to  a  fraction  whose  denominator  is  180. 


ADDITIOX    OF    FRACTIONS. 


49 


Rule. — I     Find  the  least  common  multiple  of  the  given  denom,inators. 
II.    Divide  this  inidtiple  hy  the  denmrviaator  of  each  of  the  given  frac- 
Mons,  and  multiply  its  uum^erator  hy  the  quotient  thus  obtained. 

Remarks. — 1.    The  pupil  should  do  as  much  of  this  work  as  possible  by  inspection. 
2.     Mixed  numbers  should  be  reduced  to  improper  fractions  before  applying  the  rule. 

EXAMPLES  FOR  PKACTICE. 

184.     Eeduce  to  equivalent  fractions  having  the  least  common  denominator: 

7.  23i,  14f,  7^,  5f,  and  f . 

8.  17,  2f,  14i,  8f,  3^\,  and  5. 

T5    3j  f  J  ?5  "B"'  T>    8j  TTj   ^^^  A- 


1-  h  h  h  and  f . 

2-  h  h  h  h  and  |. 
S.  h  -h,  "!>  h  ^>  and  -jL. 
4.  A,  i,  h  h  2,  H',  and  4. 
^.  h  h  tV  -^>  h  ¥.  and  5. 
■6.  A.  i^  11'  n,  n,  ^,  and  1 


9. 
10. 
11. 
12. 


A,  h  U,  \h  8,  iV  3i,  in,  and  ^. 
it,  iV  i  14, 1,  il, Vr«,  W,  5f,  11. 
il,  -H,  V,  \h  3,  2i,  4^3^,  2,  tf 


ADDITION   OF   FRACTIONS. 

185.     To  Add  Fractions  having  a  Common  Denominator. 

Example. — Find  the  sum  of  |,,  I,  |,  J,  and  |. 


OPEKATION. 


+  i  +  «9  +  i  +  t  =  V 

V  =  2i. 


Explanation. — As  the  given  fractions  have  a  com- 
mon denominator,  their  sum  may  be  found  by  adding  the 
numerators  and  placing  the  result  19,  over  the  common 
denominator  ;  the  simplest  form  of  this  sum  is  found  by 
application  of  Art.  175. 


Rule. — Add  the  numerators  and  place  the  sum  over  the  coinmon 
denominator ;  if  the  result  be  an  improper  fraction  reduce  it  to  a  whole 
or  mixed  number. 

EXAMPLES    EOK   PRACTICE. 

186.    Add 


^.  f,  f,  h  i  h  and  f 

^-  if ,  tV  f^,  H,  t\,  and  tV- 

3.  is,  e,  -h,  i-i,  H,  il,  and  H- 

^-  -h^  -h,  A.  t't,  t^,  H»  and  |4. 

5-  A,  A,  t\.  H,  a,  14,  and  f  1 


6. 
7. 
<?. 

10. 


W,  ii  fi,  1^,  \l,  \h  and  ^Sg 
T7 >  TT'  fi '  2 T  ITT'  and  -f^. 
ii,  ih,  A,  ii,  ih  ih  ii,  and  If. 
A,  ii,  "jV,  "3%,  "2??,  il,  if,  and  |f. 
i^,  A.  ii,  if,  If,  If,  ff,  and  i^ 


187.    iTo  Add  Fractions  not  having  a  Common  Denominator. 

Example. — "What  is  the  sum  of  f,  4,  and  4. 

OPERATION.  Explanation.— Since  the  given  fractions  are  not  of  the 

2    ,    B^    I    1  same  unit  value,  reduce  them  to  a  common  denominator 

'         '  (Art.  181),  and  writing  their  equivalents  below,  add  their 

f  t  "I"  fr  +  ti  ^^  f  I  =^  lf4'     numerators,  and  place  the  sum  over  the  common  denom- 
inator;  reduce  this  result  to  an  improper  fraction. 

Rule. — Reduce  the  fractions  to  a  common  denominator,  or  if  desired, 
to  their  least  common  denominator ;  add  the  resulting  numerators,  place 
the  sum  obtained  over  the  common  denominator  and  reduce  the  fraction. 
4 


50 


AUDITION'   OF    FRACTIONS. 
EXA3IPL£S  FOK   PKACTICE. 


188.    Add 
1-     h  I,  h  h  I,  \,  and  f. 
^-     H»  f^'  i'  I'  i'  ^'  and  yV 
^-     -fr.  ft.  f.  fi.  i,  i'  and  \\. 
4.     f ,  f ,  1*0,  \,  -h>  and  i 
^.     h  h  h  h  \,  -h^  and  \. 


6'  n,  i,  -fo,  h  h  h  H.  and  -^. 

-  h  h  t'«V,  iVf^  if.  i.  h  and  i. 

<^-  H.  ^>  h  tV.  iuS  I.  h  and  f 

^'>.  f,  f,  §,  iS.  A.  i.  and  ^K. 

10.  h  h  ii  I.  I.  *,  t'o,  and  WL. 


189.     To  Add  Mixed  Numbers. 

Example. — Find  the  sum  of  2^,  ^,  4,  and  f. 


OPERATION. 

2^  +  ^  +  4  +  1  = 


Explanation. — Write  the  expressions  in  a  hori- 
zontal line;  then  change  such  of  the  expressions  as 
are  in  mixed  or  entire  form  to  fractional  equivalents, 
and  place  them,  together  with  the  simple  fractions, 
in  a  line  below.  Next  find  by  inspection  that  90  is 
the  least  common  multiple  of  the  denominators,  or 
the  least  common  denominator  of  the  expressions; 
then  apply  Art.  170,  add,  and  reduce  results. 


For  convenience  the  fraction.?  may  be  written  in  a  vertical  line  and  only  the 
fractional  parts  of  the  expressions  reduced  :  then  adding  the  integers  and  the 
fractions  separately,  unite  the  results. 


OPERATION. 


2 

+     45 

4 

i     TO 

i    3C 

m 


Explanation. —  Separate,  mentally,  or  by  a  vertical  line,  the  integers 
from  the  fractions.  By  inspection  reduce  the  fractions  to  equivalents  hav- 
ing the  common  denominator  90  ;  now,  keeping  this  in  mind,  write  only 
the  numerators  45,  70,  and  36  ;  the  sum  of  these  is  151,  which  placed  over 
the  common  denominator  in  the  form  of  a  fraction,  gives  Vo"i  reducible 
to  1 J  J;  this  added  to  the  integers  gives  7f^,  the  sum  as  before  found. 


Rules. —  1-  Reduce  mixed  numbers  and  integers  to  common  fractional 
forms  and  then  to  coinmon  denominators.  Add  their  numerators,  place 
the  result  over  the  common  denoTninator  in  the  form  of  a  fraction,  and 
reduce  to  simplest  form.    Or, 

2.  Find  the  sum  of  the  integers  and  the  fractional  expressions  sep- 
arately, and  add  the  results. 


KXAMPL,ES   FOK   PRACTICE 

190.     Add 

1.  4,  4,  I,  i,  h  3,  \,  H,  and  11. 

2.  I,  2i,  \,  h  6,  h  h  ^n,  and  4^. 

3.  I,  h  ^,  n,  5,  3i,  h  n,  and  |. 
Jf.     2,  5|, -I,  n|4,  i,  14,  20iand  4. 
5.     \,  3,  64,  I,  19,  75i,  Vj,  and  -^,. 


6. 
7. 
S. 
0. 
10. 


3t,  10,  21^,  42^:,  84i,  and  168^^. 
m,  WA,  50f^,  29H,  and  86^- 
591^,  103e,  5oi,  400,  and  96^- 
10.3^^,  1191^,  2974,  and  188i|. 
33^  15,  124,  Ci,  25,  and  l^. 


Remark. — In  invoices  of  cloth,  «fec.,  account  of  fractional  parts  is  made  only  in  quarters 
and  merely  the  numerators  are  written;  as,  5*  =  5|,  3^  =  3^,  13*  =  13|,  etc. 


SUBTRACTION    OF   FRACTIONS.  51 

Miscellaneous  Examples  in  Addition  <»f  Fractions. 

EXAMPLES  FOR  MENTAL   PRACTICE. 


191. 

What  is  1 

1. 

\,h 

\,  and  1 

2. 

hh 

i,  and  f 

3. 

i,  i, 

h  and  f 

Jf. 

4       S 

t'  IT' 

f ,  and  4 

5.  f,  f ,  I,  and  |. 

^-  3j    To'    ?»    '"^"-    To* 

7-  i,  f ,  f,  f,  I,  and  H. 

'^.  i,  h  h  h  i^,  and  h 


9-  i,  h  h  h  -h,  and  H. 

■^^-  1.  i.  tV,  H'  ^,  h  and  If 

^^-  I'  h  Ih  \h  and  |3. 

1^-  i,  !,f,ii,  1,1,1,  i,  and  f. 


EXAMPLES  FOR   "WRITTEN   PRACTICE, 

192.     Add 

1.     130f,  69§,  600^\,  :3044J,  and  4G.      j  ^.     900,  47i,  Sf,  4,  29f,  06^?^,  and  4. 
^^     80t^,  2^,  5f,  17,  41^,  83^,  and  14f.    5.     16|,  33^,  66f,  88^,  100,  and  llGf. 
■3.     28A,  85-jV,  60^^,  400,  20|,  and  11.   1 6.     18f,  65|,  161f,  67f,  23^,  and  75. 

7.  The  six  fields  of  a  farm  measure  respectively,  10,  12|,  19^,  26^.^,  30^ J, 
and  2^  acres.     How  many  acres  in  the  farm  ? 

8.  Ten  sheep  weighed  as  follows  :  90yV,  HOi,  89f,  100,  106|,  101^,  96,  99, 
113f,  and  198^  lb.  respectively.     What  was  their  aggregate  weight  ? 

9.  A  farmer  sold  3G0|  pounds  of  pork,  167f  lb.  of  turkey,  241|  lb.  of 
chicken,  690{f  lb.  of  butter,  475  lb.  of  lard,  a  cow's  hide  weighing  97f  lb., 
71|-lb.  of  tallow,  and  three  quarters  of  Ijcef  weighing  respectively,  161:J-,  187-i, 
and  190  lb.     How  many  pounds  in  all  had  he  to  deliver  ? 

10.  For  341f  bushels  of  wheat  I  received  $375^, 
For  597|  bushels  of  barley  I  received  $500i, 
For  11204-  bushels  of  oats  I  received  $619f, 

For  316^  bushels  of  buckwheat  I  received  1200^^, 
For  250  bushels  of  beans  I  received  1525-j''^, 
For  1386^  bushels  of  potatoes  I  received  $755|^, 
For  1050^  bushels  of  apples  I  received  1301 -j^. 
For  630^  bushels  of  turnips  I  received  163^^^. 
How  many  bushels  did  I  sell  and  what  sum  was  received  for  all  ? 


SUBTRACTION    OF    FRACTIONS. 

193.     To  Subtract  Fractions  having  a  Common  Denominator. 

Example. — Subtract  f  from  |, 

OPERATION.  Explanation. — Since  the  fractions  have  a  common  denominator,  tiieir 

g  ^  difiFerence  may  be  found  by  taking  the  numerator  3  from  the  numerator 


l-f 


5,  and  placing  the  difference  2,  over  their  common  denominator 


Rule. — Subtract  the  numerator  of  tlie  subtrahend  from  that  of  the 
minuend,  and  place  the  difference  over  the  coimnoih  deiioDiinator. 

Remark. — A  proper  fraction  may  be  subtracted  from  1  by  writing  the  difference  between 
its  numerator  and  denominator  over  the  denominator.  Results  should  always  be  reduced  to 
their  lowest  terms.     Improper  fractions  may  be  treated  the  same  as  if  proper. 


52 


SUBTRACTION    OF    FRACTIONS. 


194. 


EXAMPUES  FOR  MKNTAL   PRACTICK. 

What  is  the  difference  between 


1.  \  and  \. 

2.  ^  and  f . 

3.  W  and  A- 


4.  \\  and  T*j.    I  r.    V  and  f .      |  76».  \\  and  y^. 

5.  I  and  \.  8.   \l  and  i^.      11.   1  and  if 

6.  1  and  i        1  V.  fa  and  if    I  12.   1  and  y^^. 


195.     Subtract 
1.     fl  from  f|. 

3.     ft.  from  ^. 


EXAMPL.KS  FOK   WRITTEN   PRACTICE. 


s. 


\^  from  1. 

•H  ^'•oni  f|. 

Kl  from  i^^. 


P. 

11. 
12. 


\\  from  ^f 
tVo  from  ifa. 
SV  from  ^i. 
II  from  SV- 


13. 

U- 
15. 
10. 


13.    SV  '»n<^  T^- 
U.    4"  and  f 
i.'J.   1  and  4^. 


ff  from  ^. 
H^fromH^. 
\WiYom\. 
yV  from  3. 


196.     To  Subtract  Fractions  not  having  a  Common  Denominator. 

Example. — From  f  take  |. 
OPERATION.  Explanation. — As  the  denominators  indicate  the  kind  of  parts,  and 

only  like  things  can  be  taken  the  one  from  tlie  other,  it  follows  that  before 
tlie  subtraction  can  be  performed,  the  fractions  must  be  reduced  to  a 


1-1  = 


\»  —  To  ^^  If'o-       common  denominator  ;  then,  the  difference  between  the  resulting  numera- 
tors, placed  over  the  common  denominator  gives  /^  as  a  result. 

^u\e.  —  Eeduce  the  given  fractions  to  equivalent  fractions  having  a 
common  denominator.  Subtract  the  numerator  of  the  subtraJiend  from 
the  mimerafor  of  the  minuend,  and  uidte  the  result  over  the  common 
denominator. 

Remark. — Improper  fractions  may  be  treated  in  like  manner. 

EXAMPIES  FOR  MENTAL   PRACTICE. 

197.     What  is  the  diflFercneo  between 
i.     i  and  f  ?  \  Jf-     i  and  \  ? 


2. 


i  and  f  ? 


3.     I  and  f  ? 


5.  y^andi? 

6.  \  and  f  ? 


9. 


f  and  J4  ? 

i  and  l"? 

HandtV? 


10. 
11. 
12. 


yV  and  I  ? 

H  and  If  ? 

M  and  \\  ? 


198.     Subtract 

1.  I  from  -J. 

2.  I  from  |. 

3.  ^j  from  4- 

4.  II  from  V- 


EXAMPLES   FOR  AVRITTEN  PRACTICE. 

Find  the  difference  between 


9. 

if  and  |. 

13. 

\\  and  i. 

10. 

V  and  1. 

u. 

1^  and  V. 

11. 

V  and  yV 

15. 

1  and  f 

12. 

U  and  V- 

16. 

1  and  |. 

Fnnn 

5.  I  take  -J. 

6.  \^  take  f . 

7.  V  take  V- 
<^.     il  take  i. 

199     To  Subtract  Mixed  Numbers. 

Example. — From  IGJ  take-  11  J. 
OPERATION.  Explanation. —  Reduce  the  fractions  to  a  common  denominator. 

16|  —  llf  =  Ob.serving  that  the  -^^  of  the  subtrahend  is  greater  than  the  y\  of  the 
minuend,  take  1  from  the  16  of  the  minuend,  reduce  it  to  twelfths  (|f), 
and  adding  it  to  the  ^\  obtain  \%  ;  from  this  take  the  -f^  and  the  fractional 
remainder  is  found  to  be  \\.  Having  taken  1  from  the  16  in  the  minuend, 
there  remains  15  from  which  to  take  the  11  of  the  subtrahend  ;  therefore 
the  integral  remainder  is  4,  and  the  entire  result  4^. 

Rem.\rk.— In  case  the  minuend  is  integral  subtract  1  and  reduce  it  to  a  fractional  form  of 
the  required  denominator. 


10,^  =  15^, 
i1t«V  =  11-/2- 


SUBTRACTION    OF   FRACTIONS. 


53 


'Rule. -^  Write  the  subtrahend  underneath  the  minuend.  Reduce  the 
fractional  paHs  to  like  denominators.  Subtract  fractional  and  integral 
parts  separately  and  unite  the  results. 

Remark.— In  case  the  lower  fraction  be  greater  than  the  upper,  take  1  from  the  upper  whole 
number,  reduce  it  and  add  to  the  upper  fraction  ;  from  this  sum  take  the  lower  fraction. 


EXAMPIiES  FOR  MENTAL.   PRACTICE. 

200.     What  is  the  difference  between 


1. 

2. 
S. 
4. 


6i  and  2^  ? 
5|  and  3i  ? 
12 1  and  34  ? 
17Hand^? 


3i  and  5^  ? 
8f  and  llyV  ? 
14f  andSlfJ? 
6  J  and  14|^  ? 


9. 
10. 
11. 
12. 


3^  and  12^  ? 
17-tV  and  b\  ? 
21iandllf  ? 
9^and23H? 


IS.   17-iVand22^? 
U.  12|and3Ji? 

15.  113f  andUi? 

16.  215 iV and  45 1? 


EXAMPLES  FOR   WRITTEN  PRACTICE. 


201.     From 

1.  4i  take  1\\. 

2.  18TVtake5f 
S.  79i  take  491^. 
■4.   104A  "  84f 


Subtract 
9J I  from  11. 
20  from  56f  ^. 

41 II  from  50|. 


Find  the  difference  between 


9. 
10. 
11. 
12. 


240f  and  89^. 
210if  and  250. 
200  and  1\^\. 
11444  <<    5^0  0 


13.  117^and5Tf. 
U.  95-1^  and  383-.V 
-?/>.   lOoOf^  and  2020|. 
10.  2016|  and  2503||. 


EXAMPLES   REQUIRING   THE   USE   OF  THE   PRECEDING   EXPLANATIONS. 


202.     From  the  sum  of 
I  and  -J  take  |. 
I  and  f  take  \\. 
f  and  f  take  \\. 
I  and  2|  take  4. 
'^  and  9 1  take  34. 
5f  and  4^  take  9. 


1. 

;■) 
^. 

3. 

Jf- 

5. 

6. 

13. 

u. 

15. 
10. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 


Take  the  sum  of  ^ 


Subtract  the  sum  of 

7.  -j\  and  y  from  14f. 

8.  8i  and  ^  from  20. 

9.  f  and  5  from  11^. 

10.  18f  and  IS^V  from  100/,. 

11.  20^V  and  1 5^^  f i"om  40-gi^. 

12.  201  ij  and  87f  from  304  i. 
and  f  from  tlie  sum  of  24-  and  |. 


Take  the  sum  of  3|  and  y  from  the  sum  of  4  and  8|. 

Take  the  sum  of  20  and  14yV  from  the  sum  of  18f  and  19^^. 

Take  tlie  sum  of  28y5^  and  GO^  from  the  sum  of  bO\  and  401. 

Take  the  sum  of  100^  and  28|  from  the  sum  of  60/^  and  GO^V 

Take  the  sum  of  13^  and  46  from  the  difference  between  125  and  1 1\. 

Take  the  sum  of  216|  and  lOl^from  the  difference  between  1000  and  875- 

Take  the  sum  of  45^  and  25^  from  the  difference  between  305f  and  425^. 

Take  the  sum  of  23  and  41/„  from  the  difference  between  21^  and  93^. 

Take  the  sum  of  91  and  5f  from  the  difference  between  lig-  and  19|. 

From  21G^  acres  of  hind,  lots  of  21  A,  IGf  A,  2Gj|-  A,  41.}  A,  and  63f 
acres  were  sold.     How  many  acres  remained  unsold  ? 

2Jf.  A  lady  went  shopping  with  135  J  in  her  purse.  She- expended  for  car  fare, 
^  of  a  doll?,r;  for  thread,  |J-  of  a  dollar;  for  gloves,  'i\  dollars;  for  a  hat,  b\  dollars; 
for  a  clock,  21|  dollars;  and  invested  the  remainder  in  linen.  How  much  was 
paid  for  the  linen? 


54  MULTIPLICATION   OP    FRACTIONS. 

25.  A  dealer  bought  u  farm  for  $3685f ,  the  crops  for  $887^,  the  stock  for 
11015^,  and  the  utensils  for  $602^.  He  sold  the  entire  i)roperty  for  $6425^%. 
Did  he  gain  or  lose  and  how  much  ? 

26.  From  the  difference  between  280^  and  1200^  take  tlie  sum  of  20^,  16|, 
5|^,  86,  UH'  it  SS  ^iid  100. 

27.  From  the  sum  of  80|,  70^,  GOf,  and  1  take  tlie  difference  between  IJ 
and  101. 

28.  From  the  sum  of  4,  f ,  ^,  \,  |,  f,  I,  |,  and  -jV  take  the  difference  between 
V  and  2. 

29.  Haying  $1302|  in  bank  I  drew  checks  for  $204|,  ^ISOi^^^,  $G40^,  1^82^, 
$20,  130^,  and  $100.     How  much  remained  to  my  credit  in  the  bank  ? 

30.  A  town  owing  $38246|,  paid,  in  '35,  $9304f ;  in  '86,  $12000^ ;  in  '87, 
14250^  ;  and  in  '88  the  remainder.     How  much  was  the  jiayment  of  1888  ? 

31.  If  I  jiay  $3500  for  a  house,  $346|  for  repairs,  $1126f  for  furniture, 
I4U0^  for  carpets  and  curtains,  and  sell  tlie  entire  property  for  $5000,  how  much 
will  I  lose  ? 


MULTIPLICATION   OF    FRACTIONS. 

203.     To  Multiply  a  Fraction  by  a  Whole  Number. 
Example. — Multiply  f  by  4. 

OPERATION   I. 

Explanation — Since  the  numerator  is  the  divi- 
3  V  4 
A  >>^  4  :— :  ^_ll__  _-  1  2  _  ^4  _  -j^i         dend,  the  fraction  may  be  multiplied  by  multiplying 

8  the  numerator  3  by  the  multiplier  4  ;  the  product  is 

^/,  which  reduced  gives  1|,  or  li.     Or,  since  the 
OPERATION  11.  denominator  8,  is  the  divisor,  the  fraction  may  be 

3  r,        1  ,  multiplied  by  dividing  this  divisor  8,  by  the  multi- 

plier 4,  which  will  give  the  reduced  form,  |  =  1^. 
This  introduces  the  principle  of  cancellation  into 
or,  -j;  X  ^  =  2  =  li'  fractional  operations. 


X  4  = =  t  =  U  ; 

8--4       * 


For  examples  containing  concrete  numbers,  reason  as  follows  : 

Example. — If  one  pound  of  wool  costs  f  of  a  dollar,  what  will  be  the  cost  of 
21  i)ounds  ? 

Explanation.— Since  one  pound  costs  |  of  a  dollar,  21  pounds  which  are  21  times  1  pound 
will  cost  21  times  |  of  a  dollar,  or  7|  dollars. 

Bule. — Multiply   the   numerator,   or  divide   the  denominator,  by   the 
whole  munber. 

Remark. — To  economize  time  and  space  divide  the  denominator  or  cancel  when  it  can  be 
done,  as  the  numbers  to  be  treated  are  tlius  put  into  simpler  form. 


MULTIPLICATION   OF    FRACTIONS. 
EXAMPLES  rOK  MENTAt    PRACTICE. 


204.     What  is  the  product  of 

1.  I  multiplied  by  3  ?     \    6. 

2.  \  multiplied  by  2  ?     |     7. 

3.  %  multiplied  by  4  ?  \  8. 
Jf.  j\  multiplied  by  d  ?  9. 
S.     -^  multiplied  by  7  ?    1  10. 


f  multiplied  by  3  ? 
\^  multiplied  by  5  ? 
I  multiplied  by  8  ? 
-j\  multiplied  by  15? 
-^^  multiplied  by  6  ? 


11. 
12. 
13. 
U- 

lo. 


205.     Multiply 

1.  ^  by  85. 

2.  41  by  8. 
^.     V  by  12. 
4.     V  by  9. 


KXAMPI.ES  FOR  WRITTEN   PRACTICE. 


6. 


3J  by  16. 

V  by  11. 

i|4  by  40. 

V  by  28. 


9. 
10. 
11. 
12. 


V/  by  10. 
\V  by  57. 
t\  by  21. 
H'  by  20. 


55 


\  multiplied  by  12  ? 
1^  multiplied  by  5  ? 
f  multiplied  by  C  ? 
y\  multiplied  by  30? 
■^  multiplied  by  32? 


13.  II  by  115. 

IJt.  \V  by  49. 

15.  ^?jijbyl05. 

16.  ^\  by  156. 


Remark. — It  is  sometimes   desirable  to  reduce  the  whole  number  to  fractional  form  by 
placing?  1  for  its  denominator. 

206.     To  Multiply  a  Whole  Number  by  a  Frattion. 
Example. — Multiply  6  by  yV 


Operation  I. 

Operation  II. 

■jiV  X  0  =  f  =  3^. 

Operation  III. 

!x^  =  |  =  3^. 

3d 


Explanations. — 1st.  If  the  multiplicand  6,  be  multiplied 
by  7,  the  numerator  of  the  fraction  yV,  the  product  42  will  be 
12  times  too  large,  because  the  multiplier  was  not  7,  but  one- 
twelfth  of  7;  hence  this  product  must  be  divided  by  12,  which 
gives  If  =  S^-V  =  3*. 

2d.     Since  6  and  -^^  are  the  factors  of  the  product,  and  as  it 
matters  not  which  term  is  multiplied,  reverse  the  order  of  the 
factors  and  proceed  as  in  Art.  203. 
Place  1  as  a  denominator  for  the  multiplicand,  theu  cancel  and  reduce. 


Rule. — Multiply  the  ivlvole  numher  hij  the  numerator  of  the  fraction, 
■and  divide  the  product  by  the  denominator.    Cancel  when  possible. 


EXAMPLES  FOR   MENTAL  PRACTICE. 


4.  9  multiplied  by  W. 

5.  14  multiplied  by  |. 


6. 


207.     What  is  the  product  of 

1.  5  multiplied  by  f. 

2.  7  multiplied  by  f 

3.  6  multiplied  by  |. 


22  multiplied  by  ^. 

7.  40  multiplied  by  f. 

8.  9  multiplied  by  |. 

9.  4  multiplied  by  \\. 
10.     7  multiplied  by  |f . 


11.  15  multiplied  by  ^. 

12.  21  multiplied  by  \\. 

13.  12  multiplied  by  \. 
H.  18  multiplied  by  |. 
15.  42  multiplied  by  4- 


208.     Multiply. 

1.  81  by  f 

2.  56  by  |. 

3.  61  by  f 
J^.     105  by  \\. 


EXAMPLES  FOR  WRITTEN  PRACTICE. 


19  by  V/. 
22  by  i||. 
240  by  \l. 
8  by  li. 


9. 
10. 
11. 
12. 


27  bv  f 
210byfi. 
48  by  H • 
91  by  i|. 


13. 

U. 
15. 
16. 


71  by  H. 
203  by  ^. 
415  by  V. 
672  by  f. 


Remark. — Entire  or  mixed  number  can  be  treated  with  facility  by  reducing  them  to  fractional 
forms  and  cancelling  when  possible. 


56 


MULTIPLICATION    OF   FRACTIONS. 


209.     To  Multiply  a  Fraction  by  a  Fraction. 
Example. — 1.     (Abstract).     Multiply  f  by  4. 

Operation.  Explanation.— The  multiplier,  2,  is  equal  to  3  times  ^ 

|X|  (3  times  ^,      or  1  of  3.     By  application  of  203,  multiply  ?  bj-  3,  which 

^  =z  -I  or  gives  ?,  -which  must  be  seven  times  too  large,  since  the 

(      1^  of  3.         multiplier -n-as  not  3,   but  one-seventh  of  3;  hence,  the 

-|  X  3  =  -|;  -1x4-  =  -^.  conect  result  will  be  obtained  by  dividing  the  product,  0, 

by  7,  which  gives  /^ . 

Example. — 2.  (Concrete).  If  a  pound  of  tea  co^s  ^  of  a  dollar,  what  will  ^ 
of  a  pound  cost  ? 

Explanation.— If  a  pound  of  tea  costs  |  of  a  dollar,  i  of  a  pound  will  cost  ^  of  |  of  a 
dollar,  or  ^V  of  a  dollar;  if  J^  of  a  pound  costs  ^\  of  a  dollar,  i  which  is  2  times  i,  must  cost  2 
times  j\  of  a  dollar,  or  -j^  of  a  dollar. 

Example. — 3.  If  a  yard  of  cloth  costs  1|-  (or  y )  dolhirs,  what  will  |  of  a  yd. 
cost  ? 

ExPLAN.^TiON. — If  1  yd.  costs  y  dollars,  ^  of  a  yd.  will  cost  ^  of  5/,  orf  of  a  dollar;  and  if 
J  of  a  yd.  costs  I  of  a  dollar,  i  which  is  4  times  |,  must  cost  4  times  f ,  or  §  =  1^  dollars. 

REM.4.RKS — 1.  Observe  that  the  numerator  of  the  product  is  the  product  of  the  numerators 
of  the  factors,  and  that  the  denominator  of  the  product  is  the  product  of  the  denominators  of 
the  factors. 

2.  This  will  apply  to  the  product  of  any  fractions,  proper  or  improper,  or  to  the  product 
of  continued  fractions. 


Rule. — I-    Cujicel   all   equivalent   factors  from  the    numerators  and 
denojtiinators. 

II.  Multiply  togetlier  the  remaining  nmnerators  for  the  numerator  of 
the  product,  and  the  reniniiiiiig  dcuomijiators  for  the  dcnomiiiator  of 
the  product. 

EXAMPLKS    rOK  MENTAL   PRACTICE   (ABSTRACT). 

210.     Find  th( 
1.     \  and  4. 
?.     -I  and  |. 
■?.     I  and  \. 
4.     \  and  |. 


luc 

t  of 

Multiply 

5. 

1  and  I 

9.    ^byf 

6. 

f  and  ■^. 

^0.  ifi^yf. 

7. 

ii  and  f . 

11.    Mhyf. 

s. 

■1  i^y  T-v- 

]2.    f3-bv|. 

IS. 

U. 

15. 

16. 


Hbyl. 


EXAMPLE.S  for  mental  practice  (CONCRETE). 

211.     1.     "What  will  be  the  cost  of  |  of  a  pound  of  tea,  if  the  cost  of  a  pound 
be  f  of  a  dollar? 

2.  I  bouglit  f  of  an  acre  of  land  and  sold  ^  of  my  purchase.     What  i)art  of  an 
acre  did  I  sell? 

3.  What  will  be  the  cost  of  -J  of  f  of  a  cord  of  wood  at  $6  per  cord? 

4.  A  girl  having  |  of  a  yard  of  ribbon  used  |  of  what  she  had.     What  part  of 
a  yard  had  she  left. 

5.  John  was  given  f  of  a  farm  and  James  |  as  much.     What  part  had  James? 

6.  If :}:  of  a  stock  were  lost  by  fire  and  the  remainder  sold  at  |  of  its  cost,  what 
part  of  the  first  cost  was  received. 


DIVISIOX    OF    FRACTIONS.  57 

7.  Divide  20  into  two  parts,  one  of  which  shall  be  f  of  the  other. 

8.  So  divide  $150  between  two  persons  that  one  may  have  \  of  the  whole  more 
than  the  other. 

9.  Tea  costing  f  of  a  dollar  per  2)ound  is  sold  for  |  of  its  cost.     For  what 
price  per  pound  is  the  tea  sold? 

10.  Coffee  costing  -^^  of  a  dollar  per  pound  is  sold  for  ^  of  its  cost.     What  price 
is  obtained  for  the  coffee? 

11.  What  is  ^  of  f  of  a  yard  of  cloth  worth,  if  the  entire  yard  costs  f  of  a 
dollar? 

12.  If  a  pound  of  steak  costs  -^-^  of  a  dollar,  what  will  \  of  ^  of  a  pound  cost? 

13.  After  paying  ^  a  dollar  for  a  pound  of  nuts,  I  sold  -|  of  |  of  my  purchase 
at  the  same  rate.     How  much  did  I  receive  ? 

H.     From  a  gallon  of  oil  f  of  |  of  a  gallon  leaked  away.     AVhut  2)urt  of  a  gallon 
was  left? 

EXAMPI.KS   FOK   IVKITTKN   PKACTICK. 

212.     Multiply  together 

1-  iil,  A,  II.  ^"tUi- 

2.  3|,  I,  I,  21,  and  llf. 

3.  I,  I,  U,  ll^V  V,  and  20. 

4.  12f,  \^,  21|,  60i,  and  fg. 

5.  5^,  8,  ll^io,  17i,  25f,  and  6. 

6.  ^,  19|,  29f,  39|,  and  49. 
7-  h  h  h  i,  h  h  I,  and  10. 


<^.  200^,  187^,  40yV,  and  51f 

9.  500,  186y\,  63f,  41,  19f,  and  4. 

10.  23^^,  16f,  8|,  and  12f 

11'  I,  1^,  li  ¥,  V.andH- 

1'-  h  h  i,  h  -^,  and  60. 

13.  13^,  21f,  12,  4,  2|,  and  3^. 

U-  -R,  if,  5.1^,  H,  audi 


15.  What  will  be  the  cost  of  74^  tons  of  hay  at  f  of  15^  dollars  i)er  ton? 

16.  Having  bought  |-  of  a  farm  of  lOG^  acres,  I  sold  f  of  my  ])urchase.     How 
many  acres  do  I  sell? 

17.  I  bought  a  house  for  $21654^  and  sold  it  for  -^1^  of  its  cost.     How  many 
dollars  did  I  lose? 

18.  If  71"  barrels  of  flour  be  consumed  by  a  family  in  ten  months,  hoAV  many 
barrels  would  fifteen  such  families  consume  in  134  months? 

19.  Having  bought  2^  tons  of  coal  at  -f  of  IGf  dollars  per  ton,  I  gave  in  i>fly- 
ment  a  twenty  dollar  bill.     How  much  change  should  I  have  received  ? 

20.  If  17^  cords  of  wood  are  bought  at  ^  of  13|  dollars  per  cord,  and  sold  at 
I"  of  9^  dollars  per  cord,  what  is  the  gain  or  loss? 


DIVISION   OF   FRACTIONS. 

213.     To  Divide  a  Fraction  by  a  Whole  Number. 

Example — 1.     Divide  -^  by  2. 

OPERATION.       Explanation.— By  the  General  Principles  of  Fractions,  dividin":  the  num- 
,       erator  (dividend)  divides  the  fraction;  hence,  divide  the  numerator  4  of  the 
*  "^         ^"     fraction  i  by  2,  and  the  quotient  is  f . 


58 


DIVISION    OF    FRACTIONS. 


Example. — •->. 
cost  ? 

OPERATION. 


If  a  jjouud  of  tea  costs  f  of  a  dollar  what  will  4  of  a  i»ound 

ExPLASATiox. — If  one  pound  costs  ^  of  a  dollar,  i  of  a  pound  will 
cost  i  of  ^,  or  f  of  a  dollar.     Observe  that  in  this  operation  the  mul- 
.  .?.  tiplier  +  is  the  reciprocal  of  the  divisor  2,  or  1  has  been  written  under 
the  divisor  as  a  denominator,  the  divisor  inverted  and  the  work  per- 
formed as  in  multiplication. 


Example. — 3.     Divide  |  by  7. 

Explanation. — ^By  the  Greneral  Principles  of  Fractions,  if  we  multiply  the  denominator  we 
divide  the  fraction.     Therefore,  |  -=-  7  =  /j. 

Rule. — Divide  the  incvi£rator  or  jnultiply  the  dejwmiiiator  hy  tlie 
whole  number. 

Remarks. — 1.  Divide  the  numerator  if  it  be  divisible,  as  the  numbers  will  thus  be  made 
smaller. 

2.  If  the  dividend  be  a  mixed  number  and  the  divisor  an  integer,  it  is  not  necessary 
to  reduce  the  dividend  to  an  improper  fraction;  divide  the  integral  part  of  the  dividend  by  the 
divisor,  and  if  there  be  a  remainder  from  such  division,  reduce  it  to  a  fraction  of  the  same 
denomination  as  the  fractional  part  of  the  dividend,  add  it  to  this  fractional  part  and  divide  as 
before  shown. 


Example 


OPERATION. 


8)2815f 


351|| 


—4.     Divide  28151  by  8. 

Explanation. — (Short  Method). — Write  as  in  Short  Division  and  divide;  8  is 
contained  in  2815j,  351  times  with  a  remainder  of  Tj^not  divided;  reduce  the  7  to 
thirds  and  to  the  result  add  the  ^  making  -/,  which  divide  by  8,  obtaining  ?,|  as 
the  fractional  part  of  the  quotient;  annex  this  to  the  integral  part  which 
gives  351|J. 


EXAJHPI.ES  FOR  MEXTAX  PKACTICE. 


211.     What  is  the  quotient  of 


4  divided  by  3  ? 
J I  divided  by  4  ? 
-,«,-  divided  by  9  ? 
^\  divided  by  8  ? 


5. 
6. 


9. 
10. 
11. 
12. 


f  divided  by  4  ? 
|4  divided  by  7? 
14  divided  by  IS  ? 
fl^  divided  by  22  ? 


H"  divided  by  5  ? 
\\  divided  by  7  ? 
J  Of  divided  by  15  ? 
II  divided  by  13  ? 

IS.     If  a  pound  of  powder  costs  f  of  a  dollar,  what  will  ^  of  a  pound  cost  ? 

IJf.,  Having  \\  of  a  yard  of  cloth,  I  divided  it  into  7  equal  pieces.  How 
much  cloth  was  there  in  each  piece  ? 

15.  If  1^  of  a  farm  be  grain  land,  and  evenly  divided  into  3  fields,  what  i)art 
of  the  farm  will  each  field  contain  ? 


EXA3IPLKS    FOK  WKITTEX   PKACTICE. 


215.     Divide 

1.     |^by9. 

5.     -A\by8. 

0. 

308  Jj-  by  40. 

13. 

205|-  by  6. 

2.     2i^  bv  17. 

6'.     21i  by  5. 

10. 

10003Vby41. 

U- 

185 1  by  9. 

s.    |Hbyi2. 

7.     41^  by  11. 

11. 

16f  by  5. 

15. 

112^  by  8 

Jf.     3ff  by  21. 

8.     2096 1  by  21. 

12. 

108^  by  25. 

16. 

321|  by  6 

DIVISION"    OF   FRACTIONS. 


59 


216.     To  Divide  a  Whole  Number  by  a  Fraction. 

Example. — Into  liow  many  pieces  f  of  a  yard  eaub,  may  5  yards  of  ribbon  be 
cut? 

OPERATION.  Explanation. — Since   5   yd.  equals  ^f  yd.  Ibey  may  be  cut  iulo    as 

o  -^  ^  =:  many  pieces,  each  containing  f  yd.  as  |  is  contained  times  in  'j^Vhich  is 

y  ^  2  _  7|^     7i  times. 

Remark. — Since  the  denominator  names  or  tells  the  kinds,  or  value  of  the  parts  taken,  when 
fractions  are  reduced  to  the  same  denomination,  or  to  equivalent  fractions  having  a  common 
denominator,  their  numerators  compare  as  whole  numbers.  We  may  consequently  ignore  the 
denominators. 


Rules. — 1.  Multiply  the  denominator  of  the  fraction  hy  the  whole 
nuinber,  and  divide  the  result  hy  the  numerator.    <^r, 

2.  Reduce  the  ivhole  number  to  a  fraction  .of  the  same  denoinination 
as  the  divisor,  and  divide  the  numerator  of  the  dividend  hy  that  of  the 
divisor. 

KXAMPLES   FOK   MKXTAL   I'KAtTICK. 


217.    Divide 

1.     17  by  f 

6.     UhjU. 

11. 

31  by  V- 

16. 

Mby|. 

2.     11  by  |. 

7.     20by^. 

12. 

50  by  V- 

17. 

30  by  ^ 

3.     20  by  if. 

8.     51  by  If. 

13. 

21  by  i 

18. 

12  by  |. 

4.     86by|. 

9.     39byH. 

U- 

60  by  |. 

19. 

33  by  H 

S.     101  b^'  ^. 

10.     25byV- 

15. 

18  by  |. 

20. 

15  by  |. 

EXAMPLES  FOR   WRITTEN  PRACTICE. 

218.  1.  If  i  of  an  acre  of  land  sell  for  45  dollars,  what  Avill  an  acre  sell  for 
at  the  same  rate  ? 

2.  A  farm  of  471  acres  is  divided  into  shares  of  94^  acres  eacli.  How  many 
shares  are  there  ? 

3.  A  church  collection  of  232  dollars  was  divided  among  poor  families  to  each 
of  which  was  given  5f  dollars.     How  many  families  shared  the  bounty  ? 

4.  When  potatoes  are  worth  f  of  a  dollar  per  bushel  and  apples  -|  of  a  dollar 
per  bushel,  how  many  bushels  of  potatoes  will  pay  for  a  load  of  apples  measuring 
30  bushels  ? 

5.  A  woman  buys  f  of  a  cord  of  wood  worth  Gf  dollars  per  cord  and  jiays  for 
it  in  work  at  i  of  a  dollar  per  day.  How  many  days  must  she  work  to  make  full 
payment  ? 

6.  A  dealer  paid  -f  of  15f  dollars  for  |  of  14:^  cords  of  wood.  "What  was  the 
cost  per  cord  ? 

7.  If -j\  of  a  farm  of  67-J-  acres  be  divided  into  63  village  lots,  what  part  of  an 
acre  will  each  lot  contain  ? 

8.  1760  bushels  of  wheat,  2100  bushels  of  barley,  2758  bushels  of  oats,  and 
696  bushels  of  beans  were  put  into  sacks  ;  those  for  the  wheat  contained  each 
2J  bushels,  for  the  barley  2|  bushels,  for  the  oats  2J  bushels,  and  for  the 
beans  1|  bushels.     How  many  sacks  in  all  were  required  ? 


60 


DIVISION   OF   FRACTIONS. 


219.     To  Divide  a  Fraction  by  a  Fraction. 

Example. — Divide  |  by  4. 

Operation. 

i  4  times  4- 
or 


(     iof4 

0  =^^- 


4  = 


Explanation. — Thedivisor,  i,  is  equal  to  4  times  |, 
or  !  of  4.  Appl3ing  the  explanation  of  Art.  213,  and 
dividing  the  dividend,  :-| ,  by  4,  gives  /o  as  a  quotient ; 
but  since  tlie  given  divisor  was  1  of  4,  and  the  divisor 
used  was  4,  a  number  7  times  too  great,  -^^j,  the  quotient 
obtained,  is  7  times  too  small;  to  correct  this  error 
multiplj'  /j  by  7,  obtaining  f  J  as  an  answer.  Ob- 
serve that  21  (the  numerator  of  the  quotient),  i» 
obUiined  by  multiplying  the  numerator  of  the  divid- 
end by  the  denominator  of  the  divisor,  and  that  20  (the  denominator  of  the  quotient),  is 
obtained  by  multiplying  the  denominator  of  the  dividend  by  the  numerator  of  the  divisor,  or 
by  effecting  a  cross  multiplication  as  shown  by  the  connecting  or  tracing  lines  in  the  operation. 

Rules. — 1.  Multiply  the  numerator  of  the  dividend  hy  the  denomina- 
tor of  the  divisor  for  the  numerator  of  the  quotient,  and  multiply  the 
denoniinator  of  the  dividend  hy  the  numerator  of  the  divisor  for  the 
denominator  of  the  quotient;  Or, 

2.  Invert  the  terms  of  the  divisor  and  proceed  as  in  multi plication  of 
fractions. 

Remark. — Reduce  mixed  numbers  to  improper  fractions  before  applying  the  rule. 

•      KXAMPLKS   FOK   MKNTAI.,   PKAOTK'K. 


220.     Divide 

1.  f  bv  |. 

2.  |byf. 
S.     fbyf 
4.     ibyf. 

5-     i  by  f . 

6.  ^jbyf. 

7.  ^byi^. 
8-  iVbyf 

^-     iibyf. 

10.  |byi 

11.  fbyii. 

12.  Ibyf. 

IS. 

u. 

15. 
16. 

\i  by  4. 
-^  by  W. 
Hbyi 
H  by  |. 

221.     Divide 

KXAMPLKS  nm.  w 

KITTKN    PKACTICK. 

1-  HbyH- 

2-  liby^f 

3-  iVVbyT^- 
4.    Hbyif. 

•^-    fUbyii- 

6-     HbyW- 
7.     Hiibyff 
S.     IfbyV.^       1 

9.      VjO  by  Yi. 

10.    y>  hi  I 

11'    iVbyV- 
12.    HbySV- 

IS. 

U- 
15. 
16. 

V  by  A- 
U-  by  1?. 
Hbyi 
It  by  ■^^. 

17.  If  11  boy  earns  ^  of  a  dollar  in  a  day,  how  long  will  it  take  him  to  earn  $15f  ? 

18.  How  many  fields  of  Oi;  acres  each  can  be  made  from  a  farm  containing 
125 1  acres  ? 

19.  If  a  wlieelman  runs  93|  miles  jter  day,  bow  long  time  will  he  require  to  run 
1167^  miles? 

20.  If  12  J  acres  produce  982^  l)ii.  of  corn,  how  many  bu.  will  15|^  acres  produce? 

21.  If  ten  men  cut  1324^  cords  of  wood  in  six  days,  how  many  cords  can  eighteen 
men  cut  in  twenty-one  days  ? 

22.  If  a  man  bought  1150J  bushels  of  wlieat  with  f  of  his  money,  how  many 
bushels  could  he  have  bought  had  all  his  money  been  invested  ? 

23.  After  traveling  ^  of  the  distance  between  two  cities,  a  pedestrian  finds 
that  there  are  1014  miles  still  before  him.     How  far  ai)art  are  the  cities  ? 


COMPLEX     FRACTIONS.  61 

COMPLEX   FRACTIONS. 

222.       A  fraction  is  complex  when  either  or  both  of  its  terms  are  fractional. 
'  Thus  -  is  a  complex  fraction  and  is  read  5  ^  f  ;  it  indicates  that  5  is  to  be 

I"  5  2 

divided  by  f .     -  is  read  |  -^  8  and  indicates  what  is  thus  expressed.     ~  is  read 

^  -=-  1^  and  indicates  Avhat  is  thus  expressed. 

Remark. — The  entire  subject  of  complex  fractions  will,  ou  account  of  its  lack  of  practical 
-value,  be  dismissed  with  the  full  illustration  of  one  example  of  each  of  two  forms. 


I 
Example  1. — What  is  the  value  of  -? 

"t 

3 

Operation:    |  =  |  -  f  =:  |  x  |  =  ti  =  ^tV- 

Example  2. — Wliat  is  the  value  of ."' 

of  ^  *       ■^' 

Operation:    ^-.-^^  =  (|  x  V)  -  (i^  X  fi)  -  |  X  ^  X  ^  X  ^  =  U^^-^,- 
i  of  i|  u 

MISCELLANEOUS   EXAMPLES   IN  FRACTIONS. 

223.     1.     From  the  sum  of  f  and  5|,  take  the  difference  betweeji  17^  and  21. 
^.     How  much  will  remain  after  the  product  of  f ,  -j^,  2,  ^^,  and  3f  is  taken 
from  10|. 

3.  Divide  into  six  equal  parts  the  product  of  ll^V  multiplied  by  3|. 

4.  Find  the  remainder  after  subtracting  the  product  of  3f,  -f,  7|,  5,  f ,  and 
1,  from  the  product  of  3,  f,  f ,  7,  f,  5,  f,  and  14. 

5.  An  estate  is  so  divided  among  A,  B,  and  C,  that  A  gets  |,  B  j^^,  and  C  the 
remainder,  which  Avas  14200.     What  was  the  amount  of  the  estate  ? 

6.  My  bank  deposit  is  $5605,  which  is  4|^  times  the  amount  in  my  purse. 
How  much  money  have  I  in  all  ? 

7.  If  14  bu.  of  apples  can  be  bought  for  $3^,  how  many  bushels  can  be  bought 
for  $!f  ? 

8.  A  woman  having  $1,  gave  f  of  it  for  coffee  at  33^^  i)er  pound.  How  many 
pounds  did  she  buy  ? 

9.  Having  bought  f  of  a  ship,  I  sold  |  of  my  share  for  $12000.  What  wjis 
the  value  of  the  ship  at  that  rate  ? 

10.  If  the  ingredients  are  -^  sulphur,  ^l.  saltpeter,  and  |  charcoal,  what 
is  the  number  of  pounds  of  each,  in  2154y\  pounds  of  guni)0wder  ? 

IJ.  AVhat  must  be  the  amount  of  an  estate  whic^h,  if  divided  into  three  ])arts, 
the  first  Avill  be  double  the  second,  the  second  double  the  tliird,  and  the  differ- 
ence between  the  second  and  the  third  be  $7500  ? 

12.  Having  paid  $115  for  a  watch  and  chain,  I  discover  that  the  cost  of  the 
chain  was  only  ^  of  the  cost  of  the  watch.     What  was  the  cost  of  each  ? 


62  MISCELLAXEOUS   EXAMPLES    IX    FKACTIOXS. 

13.  I  gave  two  20-(l()llar  gold  coins  to  :i  dealer,  of  whom  I  bouglit  2  cords 
of  wood  at  5f  dollars  per  cord,  and  3}  tons  of  coal  at  G|  dollars  per  ton.  ITow 
much  change  should  I  have  received. 

H.  A  and  B,  working  equally,  can  mow  a  meadow  in  10  days  of  9  hours  per 
day.     In  how  many  days  of  12  hours  can  A  alone  do  the  work  ? 

15.  An  estate  valued  at  ^1200()0  was  so  distributed  that  A  received  -^j  B  -^ 
of  the  estate  more  than  A,  C  as  much  as  A  and  B  together  less  $6000,  and  two 
charities  the  remainder  in  equal  j)arts.     IIow  much  did  each  charity  receive  ? 

16.  Brown  owned  -^  of  a  stock  of  goods,  ^  of  wliicli  Avas  destroyed  hy  fire  and 
^  of  the  remainder  so  damaged  by  water  that  it  was  sold  at  half  its  cost.  If  the 
uninjured  goods  when  sold  at  cost  brought  $10800,  what  must  have  been  tlie 
amount  of  Brown's  loss  ? 

17.  A  grocer  bought  a  cask  of  molasses  containing  65^  gal.,  from  which  he  sold 
at  one  time  ^  of  it,  at  another  \  of  it,  at  anotlier  5  gal.  less  than  \  of  what 
remained,  and  the  remainder  was  sold  with  the  cask  for  'Z0\  dollars.  If  the  vahie 
of  the  cask  was  one  dollar,  at  what  jirice  per  gallon  was  the  last  sale  made? 

18.  A  i^ainter  worked  17^  days,  and  after  expending  ^  of  his  wages  for  board, 
had  $15  left.     How  much  did  he  earn  per  day  ? 

19.  A  farmer  having  G50  bu.  of  wheat,  kept  for  his  own  use  52f  bu.  less  than 
\,  sold  to  his  neighbors  for  seed  454-  bu.  more  than  ^,  and  marketed  the  remain- 
der at  80^  per  bushel.     How  much  money  was  received  from  the  market  sales  ? 

20.  Of  a  journey  of  KiO  miles,  a  walker  accomplished  +  of  the  distance  the 
first  day,  \,  less  15f  miles,  the  second  day,  r^,  plus  4|  miles,  the  third  day,  and 
finished  his  journey  on  the  two  following  days  by  traveling  fifteen  hours  each 
day.     Wliat  must  luive  been  his  average  distance  per  hour  for  those  two  days  ? 

21.  A  mechanic  worked  21 1  days,  and  after  paying  his  board  with  f  of  his 
earnings,  had  66|  dollars  left.     How  much  did  he  earn  per  day  ? 

22.  So  place  a  sum  of  money  that  \  of  it  shall  be  in  the  first  package,  f  in  the 
second,  -^^  in  the  third,  and  the  remainder,  which  is  $550,  in  the  fourth  package. 
What  amount  of  money  will  be  required  ? 

23.  If  ^  the  trees  of  an  orchard  are  ai)ple,  \  peach,  ^  pear,  ^  plum,  and  the 
remaining  21  trees  cherry,  how  many  trees  in  all  ? 

2J^.  John's  weight  is  f  as  much  as  mine,  and  Ben's  is  ^  of  Jolm's.  What  is 
my  weight  if  John  is  15  pounds  heavier  than  Ben  ? 

25.  If  12  boys  earn  $54  in  a  week,  how  much  will  15  boys  earn  in  the  same 
time  at  the  same  rate  ? 

26.  A,  B,  and  C  rented  a  i)asture  for  $37.  A  put  in  3  cows  for  4  montlis,  B, 
5  for  fi  months,  and  C,  8  for  4  months.     How  much  ought  each  to  pay  ? 

27.  Henry,  when  asked  liis  age,  replied,  "  If  7^  years  be  added  to  13^^  years, 
the  sum  will  represent  ^  of  my  age."     How  old  was  he  ? 

28.  Silas,  Harvey,  and  Eobert  have  together  $2210.  Silas  has  2^  times  as 
much  as  Harvey,  who  has  |  as  much  as  I\()l)ert.     How  many  dollars  has  each  ? 

29.  Theodore's  age  is  7^  years,  and  Herbert's  9g  years  ;  three  times  the  sum 
of  their  ages  is  8  years  more  than  the  age  of  their  mother,  who  is  5^  years 
younger  than  their  father.     What  is  the  united  age  of  the  parents  ? 


MISCELLANEOUS    EXAMPLES    IN'    FHAf'TIOXS.  63 

30.  A  farmer  sold  two  cows  for  175,  receiving  for  one  only  \  as  much  as  for 
the  other.     What  was  the  price  of  each  ? 

31.  After  selling  45  turkeys,  a  dealer  had  a  of  his  stock  remaining.  IIow 
many  had  he  at  first  ? 

32.  If  8  horses  consume  4^  bushels  of  oats  in  34-  days,  how  many  bushels  will 
12  horses  consume  in  the  same  time  ? 

33.  A  and  B  can  do  a  piece  of  work  in  10  days,  which  A  alone  can  do  in  18 
days.     In  what  time  can  B  alone  do  the  work  ? 

3Ji..  John  and  Calvin  have  agreed  to  build  a  wall  for  $(80.  If  Calvin  can 
work  only  \  as  fast  as  John,  how  shall  the  money  be  divided  ? 

35.  A  flagstaff  stands  \\  of  its  length  above  and  74  ft.  below  the  surface  of 
the  ground.     "What  is  the  length  of  the  staff  ? 

36.  What  is  the  length  of  a  pole  that  stands  f  in  the  mud,  %  in  the  water,  and 
254  feet  above  the  Avater  ? 

37.  A  colt  and  cow  cost  $124.  If  the  colt  cost  $4  more  than  tliree  times  the 
cost  of  the  cow,  what  was  the  cost  of  each  ? 

38.  What  is  the  hour  Avhen  the  time  jjast  noon  equals  4  of  the  time  to  mid- 
night ? 

39.  A  tree  84  ft.  high  was  so  broken  in  a  storm  that  the  part  standing  was  f 
the  length  of  the  part  broken.     How  many  feet  were  standing  ? 

Ji-O.  A  farmer  has  \  of  his  sheep  in  one  pasture,  f  in  another,  and  the  remain- 
der of  his  flock,  72  sheep,  in  the  third  i:)asture.     How  many  sheep  had  he  ? 

J^l.  For  a  horse  and  carriage  I  paid  $540.  What  was  the  cost  of  each,  if  the 
cost  of  the  carriage  was  1^  times  the  cost  of  the  horse  ? 

JfS  Calvin  is  84  years  old,  Leo  %\  years  less  than  three  times  as  old  as  Cal- 
vin, and  John's  age  is  3  years  more  than  the  sum  of  the  ages  of  Calvin  and  Leo. 
What  is  John's  age  ? 

JfS.  Peter  can  do  a  piece  of  work  in  12  days  and  Charles  in  15  days.  How 
many  days  will  be  required  for  its  completion,  if  both  join  in  the  work  ? 

44-  Tf  A  can  do  a  i)iece  of  work  in  21  days,  B  in  18  days,  and  C  in  15  days, 
in  how  many  days  can  the  three  working  together  iierform  the  work  ? 

Remark. — In  the  above  and  similar  examples,  reason  in  general  as  follows:  If  A  can  build  a 
wall  in  4  days,  lie  can  build  \  of  it  in  1  day;  and  if  B  can  build  the  same  wall  in  5  days,  he  can 
build  \  of  it  in  1  day.  Since  in  1  day  A  can  build  \  of  the  wall,  and  B  !  of  it,  the  two  can,  if 
they  work  together,  build  in  1  day  the  sum  of  \  and  \  or  vV  of  it;  and  since  they  can  together 
do  -^Q  in  1  day,  it  will  take  them  as  many  days  to  do  the  whole  work,  or  1%,  as  4^  is  contained 
times  in  f  g,  or  2|. 

Jf5  John  and  his  father  have  joint  work,  which  they  can  do  Avorking  together 
in  25  days.  If  it  require  60  days  for  John  working  alone  to  complete  the  work, 
how  many  days  will  it  require  for  the  father  to  complete  it  ? 

Jfi  A  man  and  boy  can  in  16  days  complete  a  job  that  can  be  done  by  the  man 
alone  in  21  days.     How  long  would  it  take  the  boy  alone  to  complete  the  work? 

J^!.  Smith  said  to  Brown,  "  f  of  my  money  is  equal  to  \  of  yours,  and  the 
difference  between  your  money  and  mine  is  $30."     How  much  money  had  each? 

JfS.  Izaak  Walton  having  lost  -|  of  his  trolling  line,  added  65  ft.,  when  he 
found  it  was  just  |  of  its  original  length.     What  was  its  length  at  first  ? 


64  MISCELLANEOUS   EXAMPLES    IN'    FRACTIONS. 

^9.  A  cistern  sprung  a  leak  by  which  |  of  its  contents  ran  out,  but  during 
the  same  time  f  as  much  ran  in.     What  part  of  the  cistern  was  filled  ? 

50.  A  dog  pursuing  a  rabbit  which  has  32  rods  the  start,  runs  11  rods  while 
the  rabbit  runs  but  9.  How  far  must  the  dog  run  before  he  can  overtake  the 
rabbit  ? 

51.  A  cistern  has  two  faucets,  by  the  larger  of  which  it  can  be  emptied  in  24 
minutes,  and  by  the  smaller  in  36  minutes.  If  both  be  opened  at  once,  what 
length  of  time  will  be  required  to  empty  the  cistern  ? 

52.  Ben  and  John  bought  a  cocoanut  for  8  cents,  of  which  Ben  paid  5^  and 
John  df.  Henry  offered  8(^  for  one-third  of  the  cocoanut,  which  offer  was 
accepted,  each  taking  and  eating  one-third  of  it.  How  should  Ben  and  John 
divide  the  8^  received  from  Henry  ? 

53.  There  are  108  bu.  of  corn  in  two  bins,  and  in  one  of  tlie  bins  there  are  12 
"bushels  less  than  one-half  as  many  bushels  as  in  the  other.  How  many  bushels 
in  each  ? 

5^  At  what  time  between  one  and  two  o'clock  will  the  hour  and  minute  hands 
of  a  clock  be  together  ? 

55.  At  what  time  between  6  and  T  ? 

56.  At  what  time  between  9  and  10  ? 

57.  At  what  time  between  10  and  11  ? 

58.  Nick  bought  a  basket  of  oranges  at  the  rate  of  3  for  2  cents,  and  gained 
■60^  by  selling  them  at  the  rate  of  2  for  3  cents.     How  many  oranges  did  he  buy? 

59.  If  vou  buv  GO  lemons  at  the  rate  of  6  for  10  cents,  and  twice  as  manv  more 
at  the  rate  of  5  for  8  cents,  and  sell  the  entire  lot  at  the  rate  of  3  for  4  cents, 
■will  you  gain  or  lose,  and  how  much  ? 

60.  So  divide  $15,000  among  A,  B,  C,  and  D,  that  their  portions  shaU  be  to 
€ach  other  as  1,  2,  3,  and  4.     What  is  the  j)urtion  of  each. 

61.  I  wish  to  line  the  carjiet  of  a  room  that  is  74^  yd.  long  and  5f  yd.  wide, 
with  duck  f  of  a  yd.  wide.  How  many  yards  of  duck  will  be  required  if  it 
shrink  -^  in  length  and  -^  in  width  ? 

62.  A  and  B  are  engaged  to  perform  a  certain  work  for  $35 j^y.  It  is  sup- 
posed that  A  does  \  more  work  than  B,  and  they  are  to  be  paid  proportionally. 
How  much  should  each  receive  ? 

63.  A  tank  has  an  inlet  by  which  it  can  be  filled  in  10  hours,  and  an  outlet  by 
which  when  filled  it  can  be  emptied  in  6  hours.  If  both  inlet  and  outlet  be 
opened  when  the  tank  is  full,  in  what  time  will  it  be  emi)tied  ? 

6Jf.  A  cistern  has  two  faucets,  by  the  larger  of  which  its  contents  may  be 
emptied  in  12  minutes  and  by  the  smaller  in  15  minutes;  the  cistern  being  full, 
the  smaller  faucet  is  left  ojien  for  G  minutes,  after  which  both  are  opened.  How 
long  before  the  cistern  will  be  emptied  ? 

65.  A  man  being  asked  his  age  replied,  "  My  mother  was  born  in  1800  and  my 
father  in  1801  ;  the  sum  of  their  ages  at  the  time  of  my  birth  was  two  and  one- 
third  times  my  age  in  1846."     How  old  was  the  man  in  1880  ? 

66.  Tiiree  men  dig  a  well  for  |i3G.  A  and  B  working  together  do  ^  of  the 
work,  B  and  C  f  of  the  work,  and  A  and  C  |  of  the  work.  How  should  the 
mouev  be  divided  ? 


MISCELLANEOUS   EXAMPLES   IN   FRACTIONS.  65 

67.  Brown  and  Smith  have  joint  work  for  16  days.  In  any  given  time  Brown 
^oes  only  |  as  much  work  as  Smith.  How  many  days  would  each  working  alone 
require  to  complete  the  work  ?  If  they  work  together,  how  should  the  145  paid 
for  the  Avork  be  divided  ? 

68.  Coe,  Hall,  Tell,  and  Lee  have  a  contract  to  dig  a  ditch  which  Coe  can  dig 
in  35  days.  Hall  in  45  days,  Tell  in  50  days,  and  Lee  in  60  days.  How  lono- 
will  it  take  all  together  to  do  the  work  ?  If  $100  be  paid  for  the  work  and  all 
join  till  it  is   completed,  how  much  should  each  get  ? 

69.  A  and  B  have  joint  work  for  21  days,  but  B  can  in  a  day  do  only  f  as 
much  as  A;  after  B  has  worked  alone  for  3  days  and  A  for  5  days,  they  unite 
and  complete  the  work.  How  many  days  will  they  require  ?  If  fi75  be  paid 
ior  the  work,  what  part  of  it  should  each  receive  ? 

70.  An  estate  was  left  to  A,  B,  and  C,  so  that  A's  part  was  ^  of  the  whole 
increased  by  a  sum  equal  to  -^  of  C's  part;  B's  was  -J-  of  the  whole  increased  by  a 
.sum  equal  to  ^  of  C's  part;  and  to  C  was  given  the  remainder,  which  was  1700 
less  than  B's  share.     What  was  the  value  of  the  estate  and  of  each  one's  share  ? 

71.  Hill,  Mann,  and  Benton  have  joint  work  for  36  days,  for  which  they  are  to 
Teceive  $200,  If  Hill  can  do  only  |-  as  much  as  Mann,  and  Benton  does  twice  as 
much  as  Mann,  in  how  many  days  could  each  working  alone  complete  the  work? 

72.  How  long  would  it  take  Hill  and  Mann  ? 
7S.     How  long  would  it  take  Hill  and  Benton  ? 

74.  How  long  would  it  take  Mann  and  Benton  ? 

75.  If  all  work  together  until  the  job  is  completed,  how  should  the  money 
be  divided  ? 

76.  A,  B,  C,  and  D,  having  joint  work  for  30  days,  A  begins  and  works  alone 
ior  2  days,  when  he  is  joined  by  B;  after  the  two  have  worked  together  for  3 
"days,  they  are  joined  by  C;  the  three  work  together  for  4  days,  when  D  joins 
them,  and  all  working  together  complete  the  work.  If  A  can  do  but  \  as  much 
as  D,  B  f  as  much  as  A,  and  C  ^q-  as  much  as  B,  how  long  would  each  alone 
require  to  do  the  entire  work. 

77.  How  long  would  it  take  A  and  B  ;  A  and  C  ;  A  and  D  ? 

78.  How  long  would  it  take  A,  B,  and  C  ;  A,  C,  and  D  ;  B,  C,  and  D  ?' 

79.  How  long  after  D  began  did  it  take  for  all  to  do  it  ? 

80.  If  $300  was  paid  for  the  work  and  the  men  worked  according  to  conditions 
^iven  in  Example  76,  how  should  the  money  be  divided  ? 


fl6  DECIMALS. 


DECIMALS. 

*2*24.  A  Decimal  Fraction  or  a  Decimal  is  u  fraction  having  for  its 
denominator  ten  or  some  power  of  ten  ;  as  10,  100,  1000,  10000.  It  expresses 
one  or  more  of  the  decimal  divisions  of  a  unit. 

225.  Decimals  mav  be  expressed  in  the  same  form  as  common  fractions;  that 
is,  with  the  denominator  written.     Practically,  liowever,  this  is  never  done. 

Remark. — The  two  points  of  difference  between  common  and  decimal  fractions  are, 
1 .  The  denominator  of  a  common  fraction  is  always  written,  while  that  of  a  decimal  is  only 
indicated. 

2.  The  denominator  of  a  common  fraction  may  be  any  number,  while  that  of  a  decimal 
must  be  10  or  some  power  of  10. 

226.  The  Decimal  Point  (  .  )  is  a  period,  and  is  used  to  limit  the  value  of 
of  a  decimal  expression,  and  to  determine  the  denominator;  in  this  latter  rela- 
tion it  takes  the  place  of  the  unit  1  of  the  denominator  when  fully  written;  as, 
in  the  decimal  expression  .3.  read  3  tenths,  the  decimal  point  considered  as  1 
and  placed  before  a  cipher,  represents  the  order  of  its  units,  and  shows  that  the 
indicated  denominator  is  10. 

Remark — When  the  decimal  point  is  used  to  separate  the  integral  from  the  fractional  part 
in  mixed  decimals,  or  dollars  and  cents  in  decimal  currency,  it  is  called  a  separatrix. 

227.  Decimals  are  either  j^/wre  or  mixed. 

228.  A  Pure  Decimal  corresponds  to  a  proper  fraction,  the  value  being 
less  than  the  unit  1  ;  as,  .3,  .17,  .206,  .5191. 

A  Mixed  Decimal  corresponds  to  an  improper  fraction,  the  value  being 
greater  than  the  unit  1  ;  as,  17.4,  5.192,  32.301:. 

229.  The  Talue  of  a  Decimal  is  computed  from  the  decimal  poiiit,  and 
the  orders  have  the  same  scale  as  integers.  A  removal  of  the  decimal  point  one 
place  to  the  right,  multiplies  the  expression  ])y  ten  ;  removing  it  two  places,  by 
100  ;  three  places,  by  1000,  and  so  on.  A  removal  of  the  decimal  ])oint  one 
place  to  the  left,  divides  the  expression  by  10  :  iico  places,  by  100  ;  three  places, 
by  1000,  and  so  on. 

230.  From  the  above  it  will  be  observed  that  if  a  cipher  be  placed  between 
the  numerical  expression  of  the  decimal  and  the  point,  the  expression  being 
thereby  removed  one  place  further  from  the  point,  will  be  divided  by  10. 
But  as  the  value  of  the  decimal  expression  is  com])utccl  from  the  point  to  the 
right,  it  follows  that  one  or  more  ciphers,  placed  after  the  decimal,  will  not  alter 
its  value.  ^  is  expressed  decimally  .3  ;  a  cipher  annexed  to  the  decimal  gives 
.30  =  -^j;  tiro  ciphers  annexed  gives  .300  =  t%VV- 

By  this  it  will  be  observed  that  the  expressions,  though  unlike  in  form  are  of 
equal  value.  Each  of  the  expressions  .5,  .50,  .500,  .5000,  .50000,  .500000,  is 
equal  to  ^. 


XUMERATIOX    OF    DKCIMALS.  67 

231.  Principles. — 1.  Decimals  increase  in  value  from  right  to  left,  and 
decrease  from  left  to  right,  in  a  tenfold  ratio. 

2.  A  decimal  shotdd  contain  as  many  jjlaces  as  there  would  be  ciphers  in 
its  denominator  if  ivritten,  the  decimal  point  representing  the  unit  1  of  such 
denominator. 

S.  The  value  of  any  decimal  figure  depends  upon  its  jjlace  from  the  decimal 
point. 

Jf.  Prefixing  a  cipher  to  a  decimal  decreases  its  value  the  same  as  dividing  it 
by  ten. 

5.  Annexing  one  or  7nore  ciphers  to  a  decimal  does  not  alter  its  value. 

NUMERATION    OF    DECIMALS. 

232.  For  Notation  and  Numeration  of  Decimals  we  begin  witli  the 
decimal  point  as  a  simple  separatrix  ;  in  the  integral  expression  the  first  i)lacc  to 
the  left  is  units  (corresponding  to  the  decimal  point  in  tlie  decimal  expression), 
the  2d  tens,  the  3d  hundreds,  etc.,  while  from  the  separatrix  to  the  right  we 
have  in  order  (the  j^oint  standing  for  units),  tenths,  hundredths,  thousandths,  etc. 

233.  The  Order  of  a  Decimal  may  he  found  by  numerating  either  from 
right  to  left  or  from  left  to  right,  only  let  it  be  remembered  that  the  decimal 
point  stands  in  the  position  of  the  unit  1  of  the  decimal  denominator. 

The  order  of  a  decimal  may  usually  be  determined  by  inspection,  if  the  fact 
to  be  drawn  from  the  following  illustration  be  observed.  If  .35  be  numerated 
from  the  right  as  in  integers,  the  point  is  in  Inindreds  place;  hence,  read  35 
hundredths;  in  .1403  the  point  is  in  ten-thousands'  jilace,  read  14G3  ten-thous- 
andths; in  .014065  the  point  is  in  millions  place,  and  is  read  14065  millionths 

234.  The  value  of  a  decimal  may  be  determined  by  the  same  numeration  as 
that  employed  in  integers. 

The  relation  of  orders  in  a  mixixl  decimal  is  clearly  shown  by  the  following 

Table. 


a 
.9 

a 

03 

§ 

i 

o 
5 

O 
3 

cc 

tt-H 

3 

«-> 

fl 

5 

O 

o 

O 

X 

'I 

m 

H 

to 

s 

32 

^ 

00 

.a 
V 

«»-. 

o 

£ 

t-i 

OJ 

<) 

t^ 

o 

u 

O 

rjl 

u 

73 

3D 

a 

a 

3 

a 

o 

13 

a 

3 

s 

'o 
o 

Q 

3 

43 

^ 

^' 

■a 

_£J 

^ 

T3 

'd 

Xfi 

03 

'O  ■ 

•C! 

Oi 

OO 

I- 

S 

iO 

■^ 

eo 

c* 

<N 

eo 

1 

I 

1 

1 

1 

1 

1 

I 

\ 

• 

T 

1 

1 

ts      9      a 

HO? 

o       -^        e 


e'     K     s     E-     s 

^       .a       ^       ^'       ja       ja 

O  «C  l^  30 

1111 


The  Integral  Part.  The  Fractional  Part. 

The  above  number  is  read  111  million  111  thousand  111,  and  11  million  111 
thousand  111  hundred-millionths. 

Remark.— It  is  belter,  in  reading  mixed  decimals,  to  connect  the  integral  and  fractional 
parts  by  and;  as,  2.5  read  2  and  5  tenths;  17.016,  17  and  16  thousandths. 


68 


NOTATION   OP   DECIMALS. 


Rule. — I.  Numerate  from  the  decirtvaJ  point,  to  determine  the  denom- 
inator. 

II.  Read  the  decimal  as  a  u-lwle  iiiunber,  and  give  to  it  the  denom/- 
ination  of  the  right-hand  figure. 


EXAMI'LES  rOR  PKACTICE. 


235.     1.     Read  .297,  .1471,  .20442,  .56007. 


2 

Read  .105,  .6931,  .214698,  .40037 

55. 

3. 

Read. 19005,  .3050408,  .690004003. 

4. 

Read  .2,  .20,  .200,  .2000,  .20000,  .200000. 

5. 

Read  18.3,  29.75,  460.215,  80.03465. 

6. 

Read  270.01,  5960.030506,  8205.506007 

7. 

Read  10002.200001,  38960041.10008634] 

L. 

8. 

Read  27000.000027,  8100081. 810000S1. 

9. 

Read  1001001.1000100001,  9003009. 000009. 

10. 

Read  39864125.86954769,  919101.01919 

11. 

Read  50000000.00000050,  1000.1000. 

12. 

Read  123456.10203040506,  801.00801. 

13. 

Read  46000046.004600046. 

u. 

Read  37538651.0352615093. 

15. 

Read  45316255.83715632650. 

236.  Read  the  following  decimals: 

1. 

.206. 

9.     5320.008641. 

17. 

2000.00020002. 

2. 

1.423. 

10.     6000.58302. 

18. 

564636.002616. 

3. 

7.005. 

11.     9001.00901. 

19. 

202020.20202. 

Jh 

19.1103£ 

1. 

12.     340006.583. 

20. 

21212121.51210021. 

5. 

170.2092 

t. 

13.     75075.07507. 

21. 

30560078.0124861. 

6. 

1050.0501. 

U.     560.00020201. 

22. 

503760.2000463. 

7. 

300.003. 

lo.     53200.56931. 

23. 

37564.03060507. 

8. 

1000.0001. 

16.     214600.086005. 

2Jh 

10023580021. 1809010724. 

NOTATION   OF   DECIMALS. 

237.  The  doubt  which  often  arises  in  the  mind  of  the  pupil  as  to  hoiu  a 
decimal  should  be  written,  may  be  entirely  dispelled  by  keeping  in  mind  the 
following  facts: 

1st.  That  they  are  fractions. 

2nd.  That  both  terms  should  be  written  or  indicated. 

3rd.  That  the  denominator  of  any  decimal  (if  written)  would  be  1,  with  as 
many  ciphers  to  the  right  as  the  decimal  contains  places. 

4th.  "When  the  numerator  (or  decimal)  does  not  contain  as  many  places  as 
the  denominator  (if  written)  would  contain  ciphers,  prefix  ciphers  to  make  the 
number  of  places  equal. 


NOTATION   OF    DECIMALS.  69 

Example. — "Write  as  a  decimal  three-tenths. 

Explanation. — Observe  that  in  writing  three-tenths  as  a  common  fraction,  the  mental 
operation  is  as  follows:  after  writing  3,  the  numerator,  you  ask  yourself  3  ichat?  the  answer 
is,  3  tenths;  then  the  ten  is  written  below  as  a  denominator,  thus  obtaining  ^%.  Now  reason  in 
the  same  way  regarding  the  decimal,  and  after  writing  3,  the  numerator,  ask  yourself  3  whatf 
and  answer,  3  tenths;  and  indicate  it  by  placing  before  the  three,  a  decimal  poin^l,  which  rep- 
resents the  1  of  the  decimal  denominator;  notice  that  the  3  occupies  one  place  corresponding 
to  the  one  cipher  in  the  denominator. 

Again,  express  decimally  416  thousandths. 

Explanation. — Write  the  416  and  ask  tphat?  answer,  thousandths,  which  is  determined  by 
numerating  from  the  right;  units,  tenths,  hundredths,  and  (the  point  answering  to  the  figure  1 
of  the  denominator)  thousandths;  then  place  the  point. 

Remark. — By  extending  and  developing  this  method  of  writing  decimals,  the  pupil  can  in  a 
few  minutes  master  the  entire  matter,  so  that  he  can  write  any  decimal  as  readily  and  with  as 
great  certainty  as  if  it  were  a  whole  nlimber. 

Rule. — I.     Wi'ite  the  decimal  the  same  as  a  whole  number,  prefixing 
ciphers  when  necessary,  to  give  to  each  figure  its  true  local  value. 
II.    Place  the  decimal  point  hefore  the  left-hand  figure  of  the  decimal. 

EXAMPLES  EOK  PRACTICE. 

238.  Express  by  figures  the  following  decimals: 

1.  Twenty-six  thousandths. 

2.  Twenty-seven  hundredths 

3.  Sixteen  ten-thousandths. 
Jf..  Four  hundredths. 

5:  Twenty-tAvo  hundred-thousandths. 

6.  Five  and  seven  tenths. 

7.  Eighty-three  and  five  hundred  four  ten-thousandths. 

8.  Seven  hundred  ten  and  two  hundred  forty-three  hundred  thousandths. 

9.  Five  hundred  and  five  hundredths. 

10.  Forty-five  and  forty-six  thousandths. 

11.  One  thousand  one  and  one  hundred  ten-thousandths. 

12.  One  thousand  eight  hundred  ninety  and  ninety  thousandths. 
IS.  Eight  hundred  fifty  and  five  hundredths. 

lit.     Ten  hundred  and  ten  hundredths. 

ADDITIONAL,   EXERCISES. 

239.  Write  as  decimals 


Eleven  and  one  hundred  seven  thousandths. 

Fifteen  and  fourteen  ten-thousandths. 

Seven  hundred  twenty-six  millionths. 

Eleven  hundred  six  and  twelve  ten-thousandths. 

Sixteen  hundred  and  sixteen  hundredths. 

Ten  million  and  ten  millionths. 

Three  hundred  and  sixty-five  hundredths. 


70 


REDUCTION    OF   DECIMALS. 


8. 

9. 
10. 
11. 
12. 
13. 

U. 
I'k 

16. 
17. 
18. 
19. 

20. 
21. 
22 

23. 

25. 
26. 
27. 
28. 
29. 


Twenty-five  thousand  four  liundred  and  eleven  liundredths. 

Twenty-one  and  fifteen  thousand  fifteen  ten-millionths. 

Eighteen  tliousand  eighteen  ten  billionths. 

Five  hundred  thousandths. 

Five  hundred-thousandths. 

Xine  hundred  millionths. 

Nine  hundred-millionths. 

Fifty-four  million,  fifty-four  thousand,  fifty-four  and  fifty-four  million 

fifty  thousand  fifty-four  ten-billionths. 
One  hundred  three  thousand  five  hundred  eighty-seven  tliousandths. 
Sixty-four  thousand  sixty-four  hundredths. 

Two  million  six  hundred  four  thousand  two  hundred-thousandths. 
Xine     billion     nineteen     million     twenty-nine     thousand     thirty-nine 

millionths. 
Seventy-seven  tenths. 
Eighty-seven  thousand  one  hundredths. 
Four   hundred  seventy-nine   million  twenty  seven    thousand   four  and 

ninety-nine  thousand  four  teu-billionths. 
Seventy  trillion  and  seven  trillionths. 
Eleven  hundred  and  eleven  ten-thousandths. 
Three  thousand  one  billionths. 
One  thousand  three  millionths. 
One  hundred-thousand  eleven  ten-millionths. 
Six  hundred  five  hundred-millionths. 
Eiorhteen  hundred  uinetv  and  eighteen  hundred  ninety  hundred-billionths. 


240.     Write  as  decimals  the  following 

"•  1  0000  ou"      -'-' 

'•  100000' 

"•    1  oooo  ■ 

9  6  641^9  0  5 

"^^       1000 

^0.  W^oWo- 


7   -»-5 

-*•   10  0* 

*'•  100(57 


■*•      100* 


12. 
IS. 

U. 

15. 


3325481 
100 


15  0  0  15 
100  00  • 


16. 
17. 
18. 
19. 
20. 


3S-I00OB7 

looooooooo- 

2  19J.fiJl080  1 
10000 


1  1  0.1  1.0  01  I 
100000      ■ 


REDUCTION   OF   DECIMALS. 

241.     To  Reduce  Decimals  to  a  Common  Denominator. 

Example  1.— Keduce  .021,  .61,  .03705,  .5,  .172:2538,  to  equivalent  decimals 
having  the  least  common  denominator. 

OPERATION. 

Explanation. — Since  the  decimal  having  the  greatest  number  of  decimal 
.  JviOUOOO  places  j^  hundred-millionths  its  denominator  is  the  least  common  denominator 
.  64000000  of  the  given  expressions;  this  highest  decimal  contains  8  places,  and  by  adding  5 
.03705000  places,  or  ciphers,  to  the  first,  6  to  the  second,  3  to  the  third,  and  7  to  the  fourth. 
.50000000  a^^  "ire  reduced  to  8  places,  or  to  hundred-millionths,  which  is  the  least  common 
.17272538    denominator  of  the  given  expressions. 


REDUCTION    OF   DECIMALS.  71 

Rule.    By  annexing  ciphers  make  the  number  of  decimal  places  equal. 

Remarks— 1.  Decimals,  like  other  fractions,  can  be  neither  added  nor  subtracted  until  reduced 
to  a  common  denominator;  but  the  scale  in  decimals  being  in  the  uniform  ratio  of  ten,  it  is  only 
necessary  to  write  decimals  for  addition  or  subtraction  so  that  the  decimal  points  are  in  the 
«ame  vertical  line  ;  the  columns  will  then  be  of  the  same  orders  of  units  ;  in  other  words  the 
decimals  will  be  practically  reduced  to  a  common  denominator. 

2.  The  denominator  of  that  expression  containing  the  highest  number  of  places  is  the  least 
common  denominator  of  the  decimals;  therefore  the  least  common  denominator  may  in  all  cases 
be  determined  by  inspection,  and  decimals  reduced  to  their  least  common  denominator  by 
simply  supplying  decimal  ciphers  until  all  have  the  same  number  of  places. 

3.  In  practice,  however,  this  is  never  done,  being  rendered  unnecessary  by  observing  to 
write  decimals  so  that  the  points  stand  under  each  other. 

Example  2. — Reduce  .7,  .23,  .187G5,  and  .175  to  a  common  denominator. 

Operation.  Explanation. — As  shown  in  the  preceding  operation,  the  effect  of  reducing 

.  7  decimals  to  a  common  denominator  by  annexing  ciphers,  is  to  cause  the  decimal 

.23  points  to  fall  in  the  same  vertical  column.     Since  annexing  ciphers  to  decimals 

.18765  does  not  alter  their  value,  omit  the  ciphers  and  write  the  decimals  so  that  the 

ji*'^  points  are  in  the  same  vertical  column. 

Rule. —  Write  the  expressions  so  that  the  decimal  points  icill  stand  in 
the  same  vertical  line. 

Remark. — This  Rule  applies  equally  to  Pure  and  to  Mixed  Decimals. 

EXAMPL.ES    FOR   I'KACTICE. 

242.  1.     Eeduce  .26,  .423,  7.05,  .56931  to  their  least  common  denominator. 

2.  Reduce  21.18,  .20463,  4636.02  to  their  least  common  denominator. 

3.  Reduce  56  hundredths,  75  millionths,  3  tenths,  and  41  thousandths  to  their 
least  common  denominator. 

Jf.  Reduce  2.36,  .0005,  .1,  .62053,  and  15.2  to  their  least  common  denom- 
inator. 

5.     Reduce  19.0043,  3.87,  38.7  and  .387  to  their  least  common  denominator. 

243.  To  Reduce  Decimals  to  Common  Fractions. 

It  has  ah-eady  been  demonstrated 

1st.     That  Decimals  are  fractions. 

2d.  That  their  denominators  are  merely  indicated,  and  that  tlie  denominator 
may  be  ex})rcssed  by  writing  1,  with  as  many  ciphers  at  its  right  as  the  decimal 
contains  places. 

Example. — Reduce  .17  to  a  common  fraction. 

Operation.       Explanation.— Since  the  decimal  contains  two  places,  its  indicated  denom- 
.     .17  =  3^0^.     inator  must  be  100. 

Rule. — Omit  the  decimal  point  and  ivrite  for  a  denominator  1  with 
as  many  ciphers  as  the  decimal  contains  places. 

Remark. — Mixed  Decimals  may  be  reduced  in  a  similar  manner. 


72 


CIKCULATING    DECIMALS. 


KXAMPLES    FOR   PRACTICE. 

244. — Reduce  to  fractions  in  their  lowest  terms 


1. 

.3 

5. 

.4625 

9. 

.42504 

13. 

.114608 

2. 

.63 

6. 

.2244 

10. 

.28828 

U- 

.315264 

S. 

.105 

7. 

.18T8 

11. 

.08004 

15. 

.2000534 

4^ 

.372 

8. 

.1900 

12. 

.24042 

16. 

.983004752 

Reduce  to  an  ordinary  mixed  number 

17.  5.16  j  20.     3005.1258 

18.  13.205  j  21.     1600.0016 

19.  117.602         I  22.     1000000.00000001 

245.     To  Reduce  a  Common  Fraction  to  a  Decimal. 


23.  1234500.0012345 
2^.  6540000.0002697 
25.     188900.0000188a 


Example. — Reduce  |  to  an  equivalent  decimal. 

First  Operation.  Explaxatiox. — From  the  definition  of  decimals,  observe  that  the 

-|  X  ^  =  "iV  ^=  .6.     denominator  must  be  10  or  some  power  of  10,   and  that  f  may  be 

reduced  to  a  fraction  the  denominator  of  which  is  10  hy  multiplying  both  its  terms  by  2.     To 

change  this  fraction  to  an  equivalent  decimal,  omit  the  denominator,  and  place  a  decimal  point 

before  the  numerator. 

Second  Operation. 

5)3.0 


,6 


Explanation. — Place  a  decimal  point  and  cipher  after  the  num- 
erator 3.     This  does  not  alter  its  value,  though  in  form  it  becomes 
3.0  =  thirty  tenths;  and  since  this  numerator  is  a  dividend  and  the 
divisor  is  5,  divide  3.0  (thirty-tenths)  by  5,  and  obtain  .6  (six-tenths),  as 
a  result,  an  equivalent  in  decimal  form  as  required. 

Rule. — Place  a  deciTnaZ  point  and  ciphers  at  the  right  of  the  numerator ^ 

divide  hy  the  d  en  ami  n  at  or,  and  from  the  right  of  the  quotient  point  off  for 
decimals  as  many  places  as  there  have  heeii  ciphers  annexed. 


examples  for  practice. 


246.     Reduce  to  equivalent  decimals 


i'    tV- 

^-  H- 

11. 

H- 

16. 

H- 

21. 

ii' 

2.     H- 

'•    ^• 

12. 

■h- 

17. 

M- 

22. 

TOT 

s.    H- 

8.     i|. 

13. 

eld- 

18. 

2?(T- 

23. 

T2T 

^    -h- 

5.  tI.. 

U- 

\h- 

19. 

il\- 

24. 

n- 

^.  «. 

10.     tAu. 

15. 

Taloo- 

20. 

^' 

25. 

An- 

CIRCULATING   DECIMALS. 

247.  Certain  common  fractions,  as  ^,  i,  |,  and  fi  cannot  be  reduced  to  an 
equivalent  decimal,  because  the  denominator  (divisor)  is  not  an  exact  divisor  of 
any  power  of  10.  Sucli  expressions  cannot  be  reduced  to  exact  decimal  forms, 
and  are  termed  repeating,  or  circulating  decimals  ;  if  used  in  the  decimal  form 
they  are  followed  by  the  sign  -+-  to  indicate  inexactness.  The  repeated  part  is 
called  a  repetend ;  as,  .3333+  is  called  the  repetend  3  ;  .171717+  is  called  the 
repetend  17  ;  .206206+  is  called  the  repetend  206. 


ADDITION   OF   DECIMALS.  73 

248.  To  Express  the  Exact  Value  of  a  Repetend. 

The  exact  value  of  any  repetend  is  a  common  fraction,  the  numerator  of 
which  is  the  rej^etend  and  the  denominator  as  many  9's  as  the  repetend  contains 
places;  thus  .333+  =  -|.       .171717+  =  ^f       .206206+  =  |of. 

Rule. — TaJce  the  repetend  for  the  numerator  of  a  common  fraction,  <ind 
for  its  denominator  ivin,te  as  many  9's  as  the  repetend  has  orders  uf  units, 

EXAMPLES   FOR   PKACTICE. 

249.  Express  the  exact  value  of  the  following  repetends 


1.  .2222  + 

2.  .7777+ 


3.  .232323  + 

4.  .105105105  + 


5.  .613613613  + 

6.  .201120112011  + 


RsaiARKS.— 1.     Limiting  marks  are  sometimes  used  ;  as,  234234  ;  they  are,  however,  of  no 
importance. 

2.  In  business,  final  results  are  carried  to  three  places,  the  fourth  being  rejected  if  less  than 
one-half,  but  if  one-half  or  more  than  one-half,  1  is  added. 

3.  In  interest  rates  or  other  multipliers,  it  is  generally  safest  to  use  a  common  fractional 
equivalent. 


ADDITION  OF  DECIMALS. 

250.     Example.— Add  .7,  2.43,  .865,  11.5,  113.2075,  and  200.00165. 
Operation. 

Explanation.— Since  by  the  decimal  system  numbers  increase  ia 


.7 

"  Rfis;  ^^  °™  "S^*  *^  '^^^^  ^°  tenfold  ratio,  and  the  decimal   point 

separates  integral  from  fractional  orders,  observe  to  write  decimals 
so  that  the  x>oints  fall  in  the  same  vertical  line,  as  units  of  the  same 
order  will  thus  fall  in  the  same  column;  the  result  of  the  addition  is 
then  obtained  in  the  same  manner  as  in  simple  numbers. 


11.5 

113.2075 
200.00165 


328.70415 

Remark.— As  before  shown,  the  decimals  added  could  be  reduced  to  a  common  denomina- 
tor, but  this  being  practically  accomplished  by  the  order  in  which  they  are  written,  the  actual 
reduction  by  supplying  ciphers  is  entirely  unnecessary. 

Rule.— TFri^e  the  decimals  so  that  the  points  luill  fall  in  the  same 
vertical  line.  Add  as  in  whole  numbers,  and  place  the  point  in  the  sum, 
directly  helow  the  points  in  the  numhers  added. 

EXAMPLES  FOR  PKACTICE. 

251.     1.  Add  4,  .37,  2.46,  19.301,  and  103.21. 

2.  Add  3.04,  25.001,   .67,  .2146,  and  819.256. 

S.  Add  30.1257,  605.2146,  1000.864532,  and  16.25694. 

J^.  Add  896.111,  9530.216753,  1111.230004,  and  1100.960005. 

5.  Add  265.4203,  1129.000111,  8.005,  .0060008,  and  1200.12000014. 

6.  Add  8046.0012,  250.0000001,  311.00555,  and  81.0081001. 

7.  Add  11000.4604,    7652.0000004,  5000.500005,  and   365.50053004. 


74 


ADDITION   OF    DECIMALS. 


8.  Add  1-4.0000864,  .0096,  250.4,  700.0007,  lUOO. 00000001,  563.3001468, 
20.2001,  10000.001001  and  896.707075. 

9.  Find  the  sura  of  seventeen  and  forty-six  ten-thousandths,  eighty-three 
and  one  tliousund  four  millionths,  five  hundred  two  and  seventy-five  hundred- 
tliousandths,  three  thousand  eleven  and  three  hundred  eleven  thousandtlis,  one 
million  six  and  six  million  one  ten-thousandths. 

10.  Add  fifty-six  thousand  twelve  and  one  thousand  twenty  millionths,  six 
and  ninety-seven  million  five  billionths,  one  thousand  five  hundred  seventy-nine 
and  twenty-six  thousand  twenty-one  hundred-thousandths. 

11.  Add  one  and  one  thousandths,  ten  and  eleven  hundred-thousandths,  one 
hundred  ten  and  nine  milliontlis,  eleven  hundred  eleven  and  ninety-nine 
billionths,  one  thousand  eight  hundred  ninety  and  ninety-seven  hundred- 
billionths,  seven  millions  and  seven  hundred-thousandths. 

12.  A  farmer  having  315.625  acres  of  land,  added  at  different  times  by  pur- 
chase, 505.85  acres,  115.75  acres,  469.2  acres  and  220.9  acres,  llow  many  acres 
had  he  in  all  ? 

13.  "What  is  the  sum  of  16.5  acres,  21.125  acres,  86.06;'5  acres  111.45  acres, 
216.05  acres,  37^  acres,  426^^  acres,  80f  acres,  and  13-/^  acres  ? 

IJf.  What  is  the  number  of  bushels  in  ten  bins  of  93.625  bu.,  111.025  bu., 
306.005  bu.,  81J|  bu.,  193|  bu.,  200f  bu.,  300.0625  bu.,  125^  bu.,  250i  bu.,  and 
136^"^  bu.  respectively  ? 

15.  I  bought  ten  bales  of  cloth  as  follows  :  32J,  41||,  39-5!^,  46J,  29^,  38^^, 
431,  41-5^,  42||^,  and  40.625  yd.  respectively.     How  many  yards  in  my  purchase? 

Resiark. — In  invoices  of  goods  only  fourths  are  usually  counted,  and  these  are  written  as 
follows :  3'  =  3|,  15 '  —  \iS^,  12-  =  12f .  By  the  omission  of  the  denominator  time  is  saved. 
In  additions,  tind  the  sum  of  the  small  figures  first  as  so  many  fourths,  reduce  to  units  and 
carry  as  in  other  addition. 

,    16.     Add  21',  54=,  17',  30',  46',  61%  80',  39%  and  24\ 

17.     Add  121',  97S  46%  111%  43,  71%  86%  50',  103%  72',  71%  and  50. 

252.     To  Add  Repetends. 

Remark. — In  addition  of  repetends,  bear  in  mind  their  equivalents;  thus,  in  adding  .6-j- 
to  .3  -{-  remember  that  the  value  of  the  first  is  J,  and  of  the  second  J;  their  sum  is  %,  or  1.  In 
all  examples  in  addition  of  repetends,  before  beginning  the  operation,  continue  the  repetends  so 
that  all  have  the  same  number  of  places,  and  in  the  right-hand  column  add  each  9  as  10 


253.  Add 

. 333333 -f 

.171717  + 
.306306  + 


EXAMPLES   FOK   I'KACTICE. 


.811357  + 


.11111  + 
.<«(«<  + 
.^46207' 
.55555  + 
.33333  + 


.2222  + 
.3333  + 

.8787+ 
.0101  + 
.3467  + 


.561561561  + 
.202202202  + 
.333333333  + 
.'504300542' 
.306306306  + 


5.  Find   the    sum   of   the   following   expressions:     105.333 +,    86.1919+, 
53.103103+,  17.66+,  204.77+,  29.11+,  815.201201+  and  73.11081108 +. 

6.  Add  .66  +,  1.2121  +,  50.55  -r,  89.99  +,  2046.33  +,  38.22  +,  106.77  +  , 
1593.44  +,  11.230230  +,  528.60916091  +  and  1102.300300  +. 


MULTIPLICATION   OF    DECIMALS. 


75 


SUBTRACTION   OF    DECIMALS. 

254.     Example. — Subtract  .17  from  .50. 

Operation. 

gg  Explanation.— For    reasons  heretofore    explained,   place  the    subtrahend 

below  the  minuend,  so  that  the  decimal  points  shall  fall  in  the  same  vertical 

•  ^ '  line.     Subtract  as  in  simple  numbers,  and  place  the  paint  in  the  remainder  below 

~  the  points  in  the  terms  above. 

.30 

Remarks. — 1.  In  case  the  number  of  decimal  places  of  the  subtrahend  be  greater  than 
those  of  the  minuend,  consider  decimal  ciphers  as  annexed  to  the  minuend,  and  subtract  as 
before. 

2.     Mixed  decimals  may  be  treated  in  the  same  manner. 

Rule. —  Write  the  terms  in  decimal  order  and  suhti^act  as  in  integers, 
placing  the  -point  in  the  remainder  heloiv  the  points  in  the  other  teinrvs. 


EXAMPLES  FOR  PRACTICE, 


255.     Subtract 

1.  .573  from  .985. 

2.  .13823  from  .668. 

3.  .8627  from  1.549. 

4.  1.232  from  6.7584. 


5.  .754352  from  2.3. 

6.  46.2906  from  100.52. 

7.  3491.5  from  4246.1005. 

8.  .0001  from  10000.1. 


9. 
10. 
11. 


24.6852  from  25. 

280. Ill  from 500.000625. 

.09  from  .900. 


12.    250. 98754  from  386.245. 


MULTIPLICATION   OF   DECIMALS. 


256. 

First  Operation. 
.17  =:  -j^''^y^  (Com.  frac'l  form.) 


Example. — Multiply  .17  by  .5. 
Explanation. 


-Write  .17  as  -^^^^  and  .5  as  ^^5,  and  apply 


8i_  — 

Too  0  — 


the  rule  for  multiplication  of  common  fractions.  Multiplying 
these  fractional  equivalents,  obtain  j^§g  as  the  common  frac- 
tional expression  of  the  product;  by  Art.  245,  this  may  be 
085.  written  .085  as  the  decimal  expression  of  the  product,  or  the 
product  required. 

Remark.— Observe  that  the  denominator  of  the  product  is,  as  in  other  fractions,  the  product 
of  the  denominators  ;  also  that  the  denominator  of  the  multiplicand  contains  tiro  ciphers,  or 
two  places,  and  that  of  the  multiplier  o?j6'  cipher,  or  one  place;  these  taken  together  contain 
three  ciphers,  or  three  places,  the  same  number  of  ciphers,  or  places,  as  are  found  in  the 
product.  Then  by  applying  the  theories  of  decimals  already  explained,  the  expression  is 
changed  to  decimal  form. 

Explanation. — Write  and  multiply  the  expressions  as  in 
whole  numbers.  Since  the  numerator  is  17  hundredths  and 
the  denominator  5   tenths,    the   product  must   be  85  thou- 

sandths.     Hence,  change  the  product  85  to  85  thousandths  by 

.085  prefixing  a  cipher  and  a  decimal  point,  thus:  .085. 

'Rule,— Multiply  as  in  whole  numbers;  then,  from  the  right  of  the 
product,  point  off  for  decimals  a  number  of  places  etjual  to  the  number 
in  both  factors,  prefixing  ciphers  if  needed  to  obtain  the  rcffuired  number. 


Second  Operation. 

.17 

.5 


DIVISION    OF    DECIMALS. 


KXAMPtES  FOR   PKACTICK 

257.     Multiply 

.78  by  .7. 

.123  by  .16. 

1.45  by  .875. 

26.08  by  1.53 
IS.     1000000  by  .0000001 


9. 
10. 
11. 
13. 


25000  by  .000025. 
8.76  by.  100. 
716.0025  by  10.1006 
7000  by  .007. 
3-100000.0081  by  81.000034. 


2085.109  by  11.256. 
1000.87  by  4621.5. 
10000  by. 0001. 
.300  by. 03. 

15.  What  will  be  the  cost  of  187.0625  acres  of  land  at  $108.08  per  acre  ? 

16.  I  sold  14.4  bales  of  cloth  of  61.625  yd.  each,  at  $.60^  per  yd.  How 
much  did  I  receive  ? 

17.  What  will  be  the  cost  of  5. 75  cases  of  paper,  the  average  weight  of  which 
is  403.625  pounds,  at  $.40375  per  pound  ? 

18.  From  10.85  acres  of  wheat  a  farmer  harvested  31.875  bushels  per  acre, 
and  sold  his  crop  at  $.9725  per  bushel.     How  much  was  received  for  the  crop  ? 

Remakks.— 1.  The  contraction  of  multiplication  of  decimals  by  restricting  the  number  of 
places  to  appear  in  the  product,  is  not  deemed  of  sulficient  practical  importance  to  justify 
presentation. 

2.  As  has  been  previously  explained,  decimal  expressions,  either  pure  or  mixed,  may  be 
multiplied  by  10  or  by  any  power  of  10,  by  removing  the  point  as  many  places  to  the  right  as 
the  multiplier  contains  ciphers.  In  such  cases  annex  ciphers  to  the  multiplicand  if  there  is  not 
already  a  sufficient  number  of  decimal  places. 


DIVISION   OF    DECIMALS. 


258.     Example.— Divide  .085  by  .17. 


First  Operation. 
085  =  nriTo' 

Too  — 


1860 
6 

■nrrs-^  It:  — To— •^' 
Second  Operation. 
.17). 085  (.5 
85 
00 


Explanation.— Since  .085  =  y^^  and  .17  =  ^V,  proceed 
as  in  Division  of  Common  Fractions ;  that  is,  invert  the 
terms  of  the  divisor  and  multiply. 

Observe  now,  that  cancelling  17  and  100  from  opposite 
terms  of  the  fractional  multiplicand  and  multiplier  there  is 
left  only  the  factor  5  for  the  numerator  and  the  factor  10  for 
the  denominator  of  the  quotient,  or  the  fraction  ^  =  .5, 

Explanation. — Divide  as  in  whole  numbers.  The  divid- 
end has  3  decimal  places  ;  the  divisor  has  2  decimal  places  ; 
the  dividend  having  one  more  decimal  place  than  the  divisor, 
point  off  one  place  from  the  right  of  the  quotient. 


Remark. — It  will  be  seen  from  the  first  operation  that  the  number  of  decimal  places  of  the 
divisor  cancels,  or  offsets,  the  same  number  in  the  dividend.  If  the  number  of  places  in  the 
terms  be  equal,  it  is  ob\ious  that  the  quotient  will  be  a  whole  number. 


Rule. — I.  Wlieii  needed,  annex  ciphers  to  the  dividend  to  make  its 
places  equal  in  number  to  those  of  the  divisor- 

11.  Divide  as  in  integers,  and,  from  the  right  of  the  quotient,  point  off 
for  decimals  as  many  places  as  the  number  of  places  in  the  dividend 
exceeds  tJiose  in  the  divisor. 


DIVISION"    OF    DECIMALS. 


77 


269.  Decimals  may  be  readily  divided  if,  in  connection  with  the  above 
explanations,  attention  be  given  to  the  following 

Suggestions. — 1.  Do  not  commence  the  division  until  the  number  of 
decimal  places  in  the  dividend  is  at  least  equal  to  the  number  of  decimal  places 
in  the  divisor.     Supply  any  deficiency  in  the  dividend  by  annexing  ciphers. 

2.  If  the  divisor  and  dividend  have  the  same  number  of  decimal  places,  the 
quotient  obtained,  to  the  limit  of  the  dividend  as  given,  will  be  a  whole  number. 

3.  If  the  number  of  decimal  places  in  the  dividend  be  greater  than  the 
number  of  decimal  places  in  the  divisor,  point  off  from  the  right  of  the  quotient 
for  decimals,  a  number  of  places  equal  to  such  excess,  prefixing  ciphers  to  the 
quotient  if  necessary. 

4.  If  after  division  there  be  a  remainder,  ciphers  may  be  annexed  to  it  and 
the  division  continued  to  exactness,  or  to  the  discovery  of  a  repetend,  or  to  the 
two  or  three  places  ordinarily  demanded  in  business  computations.  All  such 
added' ciphers  should  be  considered  as  parts  of  the  dividend. 

Remarks. — 1.  Inasmuch  as  the  main  difficulty  experienced  by  pupils  with  decimals  is 
found  in  division,  and  as  that  difficulty  increases  when  the  principles  of  decimals  are  applied 
to  practice  in  percentage,  it  is  advised  that  most  thorough  and  repeated  drill  in  division  of 
decimals  be  given  to  all  grades  of  pupils  in  all  stages  of  class  work. 

2.  From  pleasant  experience  in  teaching  this  subject,  it  is  suggested  that  teii  or  more 
examples  be  grouped  as  a  single  exercise,  and  so  arranged  that  the  numerical  quotient  be  the 
same  for  all.  The  pupil  thus  relieved  from  effort  to  determine  this  feature  of  the  quotient, 
finds  the  requirement  narrowed  down  to  the  placing  of  the  decimal  point,  and  soon  fully  mas- 
ters all  difficulty. 

EXAMPLES   FOK  PRACTICE. 


260.  Divide 

1. 

.625  by  2.5. 

2. 

15.25  by  .05. 

S. 

1100  b/4.4. 

4- 

9.5  by  19. 

S. 

9.5  by  190. 

6. 

.95  by  .019. 

/v 

36.5  by  .073. 

^. 

250  by  .0625. 

(25.) 

1  -V 

-  1  =  ? 

I    -^ 

-  .1  =  ? 

1  -i 

-  .01  =  ? 

10 

-^  .1  =  ? 

10 

■^  .01  =  ? 

.1  - 

^  1  =  ? 

.1  - 

^  .1  =  ? 

.1  - 

^  .01  =  ? 

.1  -^  .001  =  ? 
10  =  ? 


9.     1750  by  .875. 

17. 

17.5  by  17500. 

10.     3.6  by  1800. 

18. 

.44  by  .00011. 

Jl.     .005  by  200. 

19. 

10006  by. 00001. 

12.     27.405  by  .00015. 

20. 

.001  by  1000. 

13.     1396. 875' by  250. 

21. 

1.6  by. 064. 

U.     131300  l)y.V25. 

6400  by  .0000016. 

15.     62.5  by  1.25. 

23. 

.0081  by  .054. 

16.     .00875  by  125. 

U. 

1860  by  .000031. 

(26.) 

[27.) 

1  -4-  10  =  ? 

.22  - 

-  11  =  ? 

1  ^  100  =1  ? 

2.2  - 

~  .011  =  ? 

.1  -^  1000  =  ? 

220 

■-  11000  =  ? 

.001  -f-  100  =  ? 

.022 

^  110  =  ? 

.0001  -^  .1  =  ? 

.00022  -=-  11000  =  ? 

100  -V-  .00001  =  ? 

2.2  - 

f-  .000011  =  ? 

1000  -^  .01  =  ? 

2200 

-^  .00011  =  ? 

.00001  ~-  1000  =  ? 

.022 

^  110000  =  ? 

10  -^  100000  =  ? 

.0000022  -=-  1100000  = 

10000  -^  .0001  =  ? 

220000  H-  .000022  =  ? 

78 


GREATEST  COMMON    DIVISOR   OF    DECIMALS. 


{29.) 
6.25  -^  2.5  =  ? 
62.5  H-  .025  =  ? 
6250  -^  .0025  =  ? 
.0625  -^  250  =  ? 
.00025  ~   .00025  =   ? 
6.25  -4-  25000  =  ? 
.0000025  -^  .00025  :=  ? 
625000  ^  .0000025  =  ? 
.0000625  ^  2500000  =  ? 
625  -f-  .0000025  =  ? 

Find  the  sum  of  the  quotients. 


(30.) 
2.5  -^  625  =  ? 
.025  -T-  62.5  =  ? 
.0025  -V-  6250  =  ? 
.00025  H-  .625  =  ? 
.000025  -^  .000625  =  ? 
.0000025  -^  62500  =  ? 
2500  -^  .0625  =  ? 
2500000  -^  .0000625  =  ? 
.00025  -4-  6250  =  ? 
.000025  --  6250000  =  ? 

Find  the  sum  of  the  quotients. 


(38.) 

1.6  -4-  2.5  =  ? 
160  -4-  .25  =  ? 
.0016  -=-  250  =  ? 
16  -f-  .00025  =  ? 
160  -4-  250000  =  ? 
16000  ^  .000025  =  ? 
.0016  -^  .00025  =  ? 
.000016  -^  2500000  =  ? 
1600  -4-  .00025  =  ? 
1600000 -4-. 00000025=? 
Find  the  sum  of  the  quotients. 

(31.) 

440  ^  1.1  =  ? 
.00044  -f-  1100  =  ? 
4400  -^  .11  =  ? 
440  -4-  .0011  =  ? 
.0044  ^  110000  =  ? 
44000000-=-  1100000=? 
4400000  -f-  .000011  =  ? 
44000  -4-  .011  =  ? 
.00000044  H-  110000  =  ? 
4400  ^  .00011  ■=  ? 
Find  the  sum  of  the  quotients. 

Remark. — Any  decimal  may  be  divided  by  1  with  any  number  of  ciphers  annexed,  as 
10,  100,  1000,  10000,  by  removing  the  decimal  point  as  many  places  to  the  left  as  the  divisor 
contains  ciphers. 


(32.) 

(33.) 

.375  -r-  1250  =  ? 

2.25  -f-  .015  =  ? 

375  -^  .0125  =  ? 

225  -^  1500  =  ? 

.0375  -4-  12.5  =  ? 

.0225  -^  150  =  ? 

37.5  -^  .000125  =  ? 

.00225  -4-  .015  =  ? 

37500  -^  .00125  =  ? 

2250  --  .0015  =  ? 

3.75  ^  1250000  =  ? 

22500  -4-  15000000  =  ? 

.00375  -^  125000  =  ? 

.000225  H-  .00015  =  ? 

.0000375  H-  .125  =  ? 

.0000225  -4-  1500000  =  ? 

3750000  -4-  .000125  =  ? 

2.25  ^  .000015  =  ? 

.000375  -4-  12500  =  ? 

22500000  -^  .00015  =  ? 

Find  the  sum  of  the  quotients. 

Find  the  sum  of  the  quotients. 

THE  GREATEST  COMMON  DIVISOR  AND  LEAST  COMMON 
MULTIPLE  OF  FRACTIONS,  COMMON  AND  DECIMAL. 

261.  All  exi^lanations  given  in  finding  either  the  Greatest  Common  Divisor 
or  Least  Common  Multiple  of  integers  apply  equally  to  fractions,  common  or 
decimal. 

262.  To  Find  the  Greatest  Common  Divisor  of  a  set  of  Common  Fractions. 
Example. — What  is  the  Greatest  Common  Divisor  of  -j,  f,  and  |  ? 

Explanation. — First  reduce  the  given  fractions  to  a  common 
denominator  and  obtain  as  a  result,  ^g,  |g,  |o  ;  then  arrange  the 
numerators  of  the  resulting  fractions  in  a  horizontal  line.  Pro- 
ceeding as  by  previous  explanations  find  the  Greatest  Common 
Divisor  of  the  numbers  to  be  5  ;  but  since  these  numbers  are 
numerators  of  fractions  whose  common  denominator  is  30,  and  30  is  the  Least  Common 
Multiple  of  this  common  denominator,  the  Greatest  Common  Divisor  of  the  given  fractions 
must  be  5  -=-  30,  or  /^  =  \.  Notice  that  the  numerator  of  the  resulting  J  is  the  Greatest  Com- 
mon Divisor  of  the  numerators,  and  that  the  denominator  6  is  the  Least  Common  Multiple  of 
the  denominators,  of  the  given  fractions. 


Operation. 
5  )  15  — 20  — 25 

3—    4—    5 


LEAST   COMMON    MULTIPLE    OF    DECIMALS.  79 

Rule. —  Write  a  fraction  the  numerator  of  which  shall  he  tJie  Greatest 
Common  Divisor  of  the  numerators  of  the  given  fractions,  and  the  denom- 
inator the  Least  Common  Multiple  of  the  denomUiators  of  the  given 
fractions. 

263.     To  Find  the  Least  Common  Multiple  of  a  set  of  Common  Fractions. 

Example.  —Find  the  Least  Common  Multiple  of  |,  f ,  and  ^V- 

Operation.  Explanation. — Reduce  the  given  fractions  to  a  common  de- 

nominator as  before,  and  obtain  fj,  fg,  \*^  ;  the  Least  Common 
'^  )  '^■^        "^^  Multiple  of  the  numerators  is  found  to  be  1800  ;  but  the  terms 

s      J        ~  I  were  not  24,  50,  and  18,  but  |J,  H,  and  J|,  and  60  is  the  Greatest 

^    "  '_  Common  Divisor  of  60,  the  common  denominator,  therefore  the 

,  i)~ .,  Least  Common  Multiple  is  not  1800,  but  '^%%^,  or  'V;  therefore  30 

is  the  Least  Common  Multiple  of  the  given  fractions.  Observe 
that  the  numerator  of  -Y  is  the  Least  Common  3Iultiple  of  the  numerators,  and  the  denom- 
inator of  the  \"  is  the  Greatest  Common  Divisor  of  the  denominators,  of  the  given  fractions. 

Rule. —  Write  a  fraction  the  numerator  of  which  shall  he  the  Least 
Common  Multi-pie  of  the  numerators  of  the  given  fractions,  and  the 
denominator  ofivhich  shall  he  the  Greatest  Common  Divisor  of  the  denom- 
inators of  the  given  fractions. 

204.     To  Find  the  Greatest  Common  Divisor  of  a  set  of  Decimal  Fractions. 
Example. — Find  the  Greatest  Common  Divisor  of  .5,  .25,  and  .375. 

Operation. 

-  \  -t^i\       .i-rv       .-,r,-  Explanation. — Reduce  the  exoressions  to  equivalents  hav- 
' mg  a  common  denominator,  obtammg  .oOO,  .2o0,  .370.     For 

-  \  -1 QQ -Q ~-  convenience  omit  the  decimal  points,  find  the  Greatest  Com- 

\ mon  Divisor  of  the  numerators,  and  obtain  125.     Since  500, 

5  \     20 10 15  ^^*^'  ^"^  375  were  not  whole  numbers,  but  .500,  .250,  and  .375, 

the  result  is  not  125,  but  .125. 

4—        3—3 

Rule. — Reduce  the  expressions  to  the  same  decimal  order,  then  icrite 
the  Greatest  Common  Divisor  of  the  expressions  us  a  whole  number,  and 
make  it  of  th  c  decimal  order  common  to  all. 

265.     To  Find  the  Least  Common  Multiple  of  a  set  of  Decimal  Fractions. 
Example.— Find  the  Least  Common  Multiple  of  .4,  .Tl,  and  .41  (J. 
Operation. 

4  )  400        <  "vO  41G  Explanation. — Reduce  to  decimals  of  the  same  order,  ob- 

,                   ^^  taining  .400,  .720,  and  .416  ;  find  the  Least  Common  Multiple  of 

^^^  ^^^  *^^  «Mmcra<o?-s.  which   is  93600.     But  .since  the  expressions 

r  \     or           ,_  .,p  were  not  integers,  but  thousandths,  the  result  is  93600  thou- 

'  ^"  sandths,  or  93.600  =  93.6,  the  Least  Common  Multiple. 

5—      9—    2G 


80  MISCELLANEOUS   EXAMPLES   IS   DECIMALS. 

Rule. — Treat  the  expressions  as  integers  and  obtain  their  Least  Com- 
rnon  Multiple;  then  make  it  of  the  same  decimal  order  as  that  one  of 
the  given  decvtnals  which  has  the  greatest  number  of  decimal  places. 

Remark. — These  illustrations,  when  presented  before  a  class,  may  properly  be  combined. 

MISCErrjVXEOUS   EXAMPLKS   IX   DECIMALS. 

266.     1.     Add  51.01,  8.1006,  67.00102,  14.5,  1750.5072003,  100.0010041. 

2.  Add  137  thousandths,  41  hundredths,  13  millionths,  5011  ten-millionths, 
608  ten-thousandths,  200600  Imndred-niillionths. 

3.  Eeduee  \^  to  a  decimal  fraction. 

4.  Reduce  .015025  to  a  common  fraction. 
0.     Divide  38.462  by  10000. 

6.  From  3006.01  take  889.01546. 

7.  From  540.123  take  the  sum  of  81.625,  126.0972,  45.001,  and  100.1002. 

8.  If  60|  bushels  of  corn  cost  $26,785,  how  much  will  17.65  bushels  cost. 

9.  Take  the  sum  of  nineteen  millionths,  five  and  two  ten-thousandths,  and 
sixty,  from  one  hundred  six  and  three  tenths. 

10.  Multiply  the  sum  of  sixty-five  and  one  hundred  seven  millionths,  by  the 
product  of  nine  hundred  millionths  and  one  hundred  twenty  and  seventeen 
hundredths. 

11.  From  one  billion  take  two  billionths. 

12.  From  six  and  fifty-hundredths  take  five  and  sixty  hundredths. 

13.  Divide  nine  hundred  sixteen  and  two  thousand  four  millionths  by  sixteen 
ten  thousandths. 

IJf..  Find  the  cost  of  11.6  bales  of  cloth,  each  bale  containing  61f  yards,  at 
$1.54  per  yard. 

15.  What  is  the  cost  of  six  barrels  of  sugar,  weighing  301,  314,  297,  309, 
313,  and  315  pounds  respectively,  at  Z\<p  per  jjonnd  ? 

16.  How  many  tons  of  phosphate,  at  $34.88  per  ton,  will  pay  for  296.48 
bushels  of  beans,  at  $1.25  per  bushel  ? 

17.  A  contractor  received  $354.06  for  excavating  a  cellar,  at  35^  per  cubic 
jard.     How  many  yards  of  earth  were  removed  ? 

18.  If  a  wheelman  travels  10.3  hours  per  day,  how  many  days  will  be 
required  for  him  to  travel  558.0025  miles,  at  the  rate  of  7.88  miles  per  hour  ? 

19.  A  teacher's  salary  is  $1500  per  annum.  If  he  pays  $650.50  for  board, 
$119.25  for  books,  $31.85  for  other  literature,  $63.40  for  charity,  $209.25  for 
clothes,  $109.90  for  traveling  expenses,  and  $41.27  for  incidental  expenses,  how 
much  of  his  salary  has  he  left  ? 

20.  I  sold  a  lumberman  381.25  pounds  of  butter  at  $.2875  per  pound, 
2468.375  pounds  of  cheese  at  $.114  per  pound,  and  2356.5  pounds  of  dressed 
beef  at  $.07^  per  pound,  and  received  pay  in  lumber  at  $23  .125  per  thousand  feet. 
How  many  thousand  feet  of  lumber  should  I  have  received  ? 


UNITED    STATES    MONEY.  81 


UNITED    STATES    MONEY. 

267.  United  States  Money  is  the  legal  currency  of  the  United  States, 
■adopted  in  1786  and  changed  by  various  Acts  of  Congress  since  tliat  date  ;  it  is 
sometimes  called  Federal  Money. 

268.  Money  is  the  measure  of  value. 

269.  Legal  Tender  is  the  term  applied  to  such  money  as  may  he  legally 
•offered  in  the  payment  of  debts. 

270.  Bullion  is  pure  gold  or  silver  in  bars,  or  ingots,  and  "  l>ullion  value  "  is 
the  value  of  such  metal,  which  varies  from  coin  value  only  by  the  charges  for 
•coinage  made  by  the  mint. 

271.  Coin  is  the  standard  money  of  tlie  mints,  its  value  being  established 
Tiy  law. 

272.  Currency  is  coin,  treasury  notes,  bank-bills,  or  any  substitute  for 
money,  in  circulation  as  a  medium  of  trade. 

273.  A  Decimal  Currency  is  a  currency  wliose  denominations  increase 
and  decrease  by  the  decimal  scale.      United  States  money  is  a  decmial  currency. 

274.  The  Dollar  is  the  unit  of  United  States  money.  Dollars  are  written 
:as  integers,  with  the  sign  ($)  prefixed  ;  the  lower  denominations  arc  written  as 
decimals,  dimes  being  tenths,  cents  hundredths,  and  mills  thousandths  of  a 
•dollar.     Thus,  15  dollars,  1  dime,  5  cents,  5  mills,  is  written  ^15.150. 

In  business  records  and  papers,  cents  are  often  written  as  fractions  of  a  dollar  ; 
the  half-cent  is  expressed  either  as  a  fraction  (4),  or  as  5  mills.  Thus,  Sln.tS 
may  be  written  $15jVo;  '^^  cents,  1.124,  or  1.125. 

275.  The  denominations  and  scale  of  United  States  money  are  shown  in  tlie 
following 

Table. 

10  mills  =  1  cent  (c.  or  et.).  10  dimes  =  1  dollar  ($). 

10  cents  =  1  dime  (d.).  10  dollars  =  1  eagle  (E.). 

Scale.— Descending,  10,  10,  10,  10.     Ascending,  10,  10,  10,  10. 

Remarks.— 1.  The  scale  being  a  decimal  one,  all  operations  in  United  Ctatcs  money  are 
performed  the  same  as  with  common  decimal  expressions. 

2.     The  Dime  is  a  coin,  but  its  name  is  never  used  in  reading  United  Slates  money.      The 
Mill  is  not  coined;  it  is  used  only  as  a  decimal  of  the  cent,  which  is  the  smallest  money  of  the 
jnint  and  the  smallest  recognized  in  business. 
6 


S2 


U>ilTEl>    STATES    MONEY, 


Obverse. 


Obverse.  Keverse. 


Coins  of  the  United  States. 


UNITED    STATES    MOXEY.  '  83 

UNITED   STATES  COINS. 

276.  The  Coins  of  tlie  United  States,  authorized  Ijy  various  Acts  of  Con- 
gress, are  of  gold,  silver,  copper-nickel,  and  bronze. 

277.  The  Gold  Coins  of  the  United  States  arc  as  follows  . 

1.  The  Dotible  Eofjle;  value,  $20  ;  weight,  510  Troy  grains. 

2.  The  Eagle;  value,  110  ;  Aveight,  258  Troy  grains. 

3.  The  Half-Eagle;  value,  $5  ;  weight,  129  Troy  grains. 

Jf..     The  Three  Dollar  piece  ;  value,  $3  ;  weight,  77. -i  Troy  grains. 

5.  The  Quarter- Eagle;  value,  $2.50;  weight,  64.5  Troy  grains. 

6.  _  The  One  Dollar  piece  ;  value,  $1  ;  weight,  25.8  Troy  grains. 

Remarks. — 1.  All  United  States  gold  coins  are  made  of -j'jj  pure  gold,  and  j'jy  alloy  of  copper 
and  silver,  the  alloy  being  used  to  toughen  the  metal  so  as  to  reduce  the  loss  from  abrasion. 
The  alloy  used  is  never  more  than  j'„  part  silver. 

2.  United  States  gold  coins  of  standard  weight  are  legal  tender  for  all  debt^. 

278.  The  Silver  Coins  of  the  United  States  are  as  follows  : 

1.  The  Dollar;  value,  $1.00  ;  weight,  412.5  Troy  grains. 

2.  The   Half -Dollar;  value,  50^/;  weight,  192. 9  Troy  grains. 
■3.  The  Quarter- Dollar;  value,  25^v  weight,  9(i.45  Troy  grains. 
Jf.  The  Divie;  value,  10/' ;  weight,  38.58  Troy  grains. 

Remarks.— 1.  The  value  of  gold  and  silver  coins  is  based  mainly  on  their  weight  and  fine- 
ness, or  the  amount  of  pure  metal  used.  Silver  coins  are  made  of  ^'^  pure  silver  and  -i\  alloy 
of  copper. 

2.  United  States  silver  dollars  are  lef/al  tender  for  all  sums  not  otherwise  provided  for  by 
contract.     The  smaller  silver  coins  are  legal  tender  for  all  sums  not  exceeding  ten  dollars. 

279.  The  Copper-Mckel  Coins  of  the  United  States  are  as  follows  : 

1.  The  Five- Cent  incce,  called  tlie  nickel;  weight,  77.16  Troy  grains. 

2.  The  Three-Cent  piece  ;  weight,  30  Troy  grains. 

Remark.— The  ^(j-  and  3^  coins  are  composed  of  J  copper  and  J  Nickel. 

280.  TIk'  Bronze  Coin. — The  only  bronze  coin  now  issued  from  the  mint 
is  the  one  cent  piece,  weighing  48  Troy  grains,  and  composed  of  -f^^  copper  and 
j5_  tin  and  zinc. 

Remark.— The  5f/  and  3^  nickel  coins,  and  the  If  bronze  coin,  are  called  minor  coins  ;  and 
while  they  are  legal  tender  for  all  sums  not  exceeding  twenty-live  cents,  their  value  is  not  a 
bullion  value,  as  in  case  of  coins  of  gold  and  silver,  but  an  arbitrary  value  fixed  for  commercial 
convenience. 

UNITED   STATES    PAPER    MONEY. 

281.  The  Paper  Money  of  the  United  States  consists  of  Treasury  Xotes, 
Treasiirif  Certifiratrs.  and  Xational  Bauh  Bills. 

282.  Silver  Certificates.— Any  holder  of  silver  dollars,  to  the  amount  of 
ten  dollars  or  more,  may  deposit  the  same  with  the  Treasurer  or  Assistant 
Treasurer  of  the  United  States  and  obtain  therefor  Silver  Certificates,  which  are 
receivable  for  duties,  taxes,  and  all  public  debts ;  and  any  holder  of  the  smaller 
silver  coins  to  the  amount  of  twenty  dollars,  or  any  multiple  thereof,  may  obtain 
therefor  lawful  money  at  the  office  of  the  Treasurer  or  of  any  Assistant  Treasurer. 


84  BEDUCTIOX   OF   UiJTrED   STATES   MONET. 

283.  United  States  Treasury  Notes. — Treasury  Notes,  or  Greenbacks, 
are  in  the  same  denominations  as  the  Bills  of  National  Banks,  with  the  addition 
of  those  of  $5,000  and  $10,000  value  respectively.  Tiiey  are  legal  tender  for  all 
debts  except  customs  or  duties,  and  interest  on  the  public  debt,  and  are  usually 
receivable  for  these  also,  being  convertible  into  coin  on  demand  when  presented 
in  sums  of  fifty  dollars  or  more. 

284.  National  Bank  Bills. — National  Bank  Bills  are  the  notes  issued  by 
National  Banks,  under  the  supervision  of  Government,  and  these  bills  are  in 
denominations  of  $1,  $2,  $5,  $10,  $20,  $50,  $100,  $500,  and  $1000,  and  being 
secured  by  deposits  of  Government  Bonds  with  the  United  States  Treasurer,  and 
redeemable  on  demand  with  lawful  money,  are  usually  received  for  all  dues,  but 
yet  are  not  legal  tender  ;  and  a  debt  cannot  be  paid  with  these  notes  if  the  cred- 
itor states  as  his  reason  for  their  rejection  that  they  are  not  lawful  money. 

REDUCTION  OF  UNITED  STATES  MONEY. 

285.  To  Reduce  Dollars  to  Cents. 
Example. — Reduce  5  dollars  to  cents. 

Explanation. — Since  there  are  100  cents  in  1  dollar,  in  5  dollars  there  are  5  times  100  cents, 
•or  500  cents. 

Rule. — Add  two  ciphers  to  the  dollars. 

286.  To  Reduce  Cents  to  Dollars. 
Example. — Eeduce  1500  cents  to  dollars. 

Explanation. — Since  100  cents  make  1  doUar,  there  are  as  many  dollars  in  1500  cents  as 
100  cents  is  contained  times  in  1500  cents,  or  15  times,  equal  to  15  dollars. 

Rule. — Divide  the  cents  hy  100,  hy  pointing  off  two  places  from  the 
right. 

KXA3LPLES   FOK   PRACTICE. 

287.  Reduce 

1.  6  dollars  to  cents. 

2.  Ill  dollars  to  cents. 

3.  241  cents  to  dollars. 
4-  1044  cents  to  dollars. 


5.  21468  cents  to  dollars.  ;  9.  $100.98  to  cents. 

6.  1800  cents  to  dollars.    \  10.  $o.T5  to  cents. 

7.  51000  cents  to  dollars.  11.  $26.53  to  cents. 
S.  9876  cents  to  dollars.    |  U.  $157.32  to  cents. 


ADDITION   AND    SUBTRACTION    OF    UNITED   STATES 

MONEY. 

288.  To  Add  or  Subtract  United  States  Money. 

Rule. —  Write  dollars  under  dollars  and  cents  under  cents;  then  add 
or  subtract  as  in  simple  ninnhers. 

EX.\MPLES  FOR   PRACTICE. 

289.  1.  Add  ten  dollars  twenty  cents,  six  dollars  forty-eight  cents,  fourteen 
dollars  twenty-six  cents,  eleven  dollars  eighty  cents,  and  forty-six  dollars  ten 
cents. 


MULTIPLICATIOK"    OF    UNITED    STATES    MONEY.  85 

2.  Subtract  seven  hundred  sixty-five  dollars  nineteen  cents  from  nine  hun- 
dred ten  dollars  eight  cents. 

3.  A  farmer  sold  produce  as  follows  :  wheat,  for  |i761.25;  oats,  |i38:i.40; 
barley,  $816.09  ;  buckwli£>at,  |;186;92;  corn,  $1127.50;  potatoes,  $063.11  ;  hay, 
$400.50.     What  were  his  entire  sales  ? 

Jf.  A  lady  bought  groceries  to  the  amount  of  $6.85;  meats,  $2.11;  dry  goods, 
$31.75;  carpets,  $167.25;  millinery,  $13.57.  AVhat  was  the  total  amount  of 
her  purchases  ? 

5.  A  student  expends  for  tuition  and  supplies,  $118.75;  for  board,  $167.50  ; 
for  clothes,  $57.25  ;  for  entertainment  and  church,  $28.42  ;  for  charity,  $6.15. 
What  amount  does  he  ex])end  ? 

6.  The  ex])enses  of  my  house  are  as  folloAvs  :  for  interest,  $167.50  ;  taxes, 
$103.29;  repairs,  $56.82;  insurance,  $11.35;  water  rent,  $11.25;  and  gas,  $27.08. 
What  are  my  total  expenses  ? 

Remarks. — 1.  Under  some  circumstances  it  is  desirable  to  write  United  States  money, 
expressed  in  dollars  ami  cents,  without  the  $  sign  and  the  decimal  point,  with  the  decimal  part 
placed  slightly  above  that  expressing  the  integers  or  dollars ;  as  $5.25  may  be  written  5-^  ; 
thirteen  dollars  and  eight  cents  may  be  written  13""*.  This  is  advisable  only  where  the  sum 
of  several  items  is  to  be  found  by  horizontal  addition. 

2.  The  amoimt  in  each  of  the  following  examples  is  to  be  found  by  horizontal  addition. 

7.     Add  15^«,  29^  1146^  1079«,  9^3,  81*5,  12392,  601,  IS^o,  and  ll^. 
S.     Add  34650,  291^5,  lOO^i,  269ii,  and  8093. 

0.     What  is  the  sum  of  21658*,  7243^  9920^  117«S  5005o,  and  1127i*  ? 
IQ.     What  is  the  sum  of  667* S  328io,  97' ^^  goO,  20",  155i«,  1101,  28^3, 
and  6759  ? 

11.  My  bills  for  a  year  are:  for  groceries,  283^1;  meats,  135^1;  miller's  i)ro- 
ducts,  76'' 5;  coal,  412  0;  kindling,  45";  milk,  47^5;  servant,  217;  incidentals, 
915*.     What  are  my  expenses  ? 

12.  A  merchant  bought  cottons,  for  3467-5;  linens,  for  1326^5;  woolens,  for 
4215''  5 ;  delaines,  for  1025*  5  •  brocades,  for  11275 ».  If  all  were  sold  for  132562  6, 
how  much  was  gained  ? 

MULTIPLICATION   OF    UNITED   STATES   MONEY. 

290.  To  Multiply  United  States  Money. 

Rule. — Multiply  as  in  abstract  decimals. 

Remark. — Money  is  a  concrete  expression  ;  therefore  in  critical  analysis  of  its  multiplica- 
tion, the  money  cost  or  price  of  an  article  is  a  concrete  multiplicand,  the  number  of  things 
bought  or  sold  is  an  abstract  multiplier,  and  their  product  is  concrete  and  of  the  denomination 
of  the  multiplicand.  But  since  the  money  scale  is  decimal,  these  terms  may  be  interchanged 
for  convenience. 

exampi.es  for  practice. 

291.  1.  AVhat  will  be  the  amount  of  the  following  purchases  :  147f  cd. 
hard  wood,  at  $5.75  per  cd.;  206f  cd.  soft  wood,  at  $4.25  per  cd.;  4  car  loads 
slab  wood,  each  containg  16f  cd.,  at  $2. 75  per  cd.;  816^  tons  hard  coal,  at  $5,15 
ton;  and  536^  tons  soft  coal,  at  $3.85  per  ton  ? 


86  DIVISION"    OF    UNITED   STATES    MONEY. 

2.  Bought  ilb  bar.  superfiue  flour,  at  $4.85  per  bar.;  355  bar.  extra  flour  at 
$5. 15  per  bar. ;  132  bar.  rye  flour,  at  $4.90  per  bar. ;  210  bar.  corn  meal,  at  $3.70 
per  bar. ;  and  642  sacks  graham  flour,  at  88^  per  sack.     What  was  the  total  cost  ? 

S.  A  retailer  bought  35  overcoats,  at  $0.75  each;  IGO  black  suits,  at  $17.25 
each;  125  plaid  suits,  at  $14.05  each;  84  jean  suits,  at  $6.90  each;  and  50  pairs 
trousers,  at  $3.15  each.     Find  the  total  cost. 

4.  An  invoice  of  six  pieces  of  gingham  of  51^,  49^,  50*,  54*,  49-,  and  51^  yd. 
respectively,  was  sold  at  $.09|  per  yd.     What  was  the  amount  of  the  sale  ? 

5.  Six  men  worked  19f  days  each,  at  $1.90  per  day;  24:^  days  each,  at  $1.80, 
11^  days  each,  at  $1.65:  and  31f  days  each,  at  $1.25.  How  much  was  earned  by 
all  in  the  entire  time  ? 

6*.  A  laborer  received  $184.55  as  a  balance  due  him  for  his  season's  work. 
He  paid  a  debt  of  $19.25;  bought  8^  yd.  cloth,  at  $1.25  per  yd. ;  2  suits  of  clothes, 
at  $13.25  per  suit;  hosiery  and  gloves  for  $2.85;  4f  tons  coal,  at  $5.65  per  ton; 
2  cd.  wood,  at  $3.90  i)er  cd. ;  3  bar.  flour,  at  $4.75  jier  bar. ;  628  pounds  of  pork, 
at  6f^  per  lb. :  and  loaned  the  remainder  of  his  money.     How  much  did  he  loan  ? 


DIVISION   OF    UNITED   STATES   MONEY. 
•>9-2.     To  Divide  United  States  Money. 
Rule. — Divide  as  in  abstract  decimals. 

EXAMPLES   FOK   PRACTICE. 

293.  i.  If  $11421.75  be  divided  equally  among  hve  persons,  what  will  be 
the  share  of  each  ? 

2.  B  sold  18T^  acres  of  land  at  $105.25  per  acre,  and  divided  the  proceeds 
equallv  among  fifteen  persons.     What  sum  did  each  receive  ? 

3.  A  charitable  farmer  gave  15f  bushels  of  apples  worth  $.50  per  bu.,  21| 
bushels  of  potatoes  worth  $  .75  per  bu.,  and  30  bushels  of  turnips  worth  $.624^  per 
bu..  in  equal  shares  to  six  families.     What  was  the  value  of  each  share  ? 

j^.  A  dealer  bought  wheat  at  $.95  per  bu.,  oats  at  $.45  per  bu.,  and  corn  at 
$.65  per  bu.  He  paid  $332.50  for  the  wheat,  $191.25  for  the  oats,  and  $113.75 
for  the  corn.     How  many  bushels  did  he  buy  in  all  ? 

o.  C  invested  $9659.50  in  coal,  at  $5.85  per  ton;  $2645.30  in  sand,  at  $2.80  per 
cubic  vd. ;  $058.40  in  lime,  at  $1.60  per  barrel.  If  he  sold  the  coal  at  $6.05  per 
ton,  the  sand  at  $2.75  per  cubic  yd.,  and  the  lime  at  $1.75  per  barrel,  what  was 
the  gain  or  loss  ? 

6.  Having  sold  my  mill  for  $17250,  and  316  barrels  of  flour  in  stock  at 
$5.15  per  barrel,  I  invested  of  the  proceeds,  $1185.85  in  furnishing  a  house, 
$1259.30  in  utensils,  $1582.25  in  live  stock,  and  with  the  remainder  paid  in  full 
for  a  farm  of  163  acres.     Wiiat  was  the  cost  of  the  farm  i)er  acre  ? 

Rem.\rk. — In  case  exact  quotients  are  not  obtained  in  division  of  dollars,  add  two  decimal 
ciphers  and  continue  the  quotient  to  cents  ;  if  not  then  exact  add  one  cent  if  the  mills  be  5  or 
more,  but  if  less  than  5,  reject  the  mills. 


ANALYSIS.  87 


ANALYSIS. 

294.  Arithmetical  Analysis  is  the  process  of  solving  problems  inde- 
pendently of  set  rules,  l)v  deducing,  from  the  terms  stated,  the  conditions  and 
relations  required  in  their  solution. 

Remark. — The  general  subject  of  Analysis  will  be  treated  only  as  auxiliary  to  the  subject 
of  Common  Fractions,  and  the  Special  Applications  of  the  Fundamental  Rules. 

Example  1.     If  5  men  earn  %'60  m  4  days,  how  many  dollars  will  T  men  i-arn 

in  9  days  ? 

First  Explanation  (^extended). — If  5  men  earn  $30  in  4  days,  1  man,  or  1  of  5  men  will 
cam  in  4  days  1  of  $30,  or  $6  ;  and  if  1  man  earns  $6  in  4  days,  in  1  day,  which  is  \  of  4 
days,  he  will  earn  \  of  $G,  or  $li.  Then,  since  1  man  in  1  day  earns  $li,  in  9  days,  which 
are  9  times  1  day,  he  will  earn  9  times  $li,  or  $13^  :  and  if  1  man  in  9  daj's  earns  $131,  7  men, 
which  are  7  times  1  man,  will  earn  7  times  $131,  or  $941. 

Second  Explanation  (abbreviated). — If  5  men  earn  $30  in  4  days,  they  will  earn  $7i  in  1 
day  ;  and  if  5  men  earn  $7^  in  1  day,  1  man  will  earn  !  of  $7|,  or  $U  ;  since  1  man  in  1  day 
earns  $U,  7  men  in  1  day  will  earn  7  times  $li,  or  $10^  ;  and  if  7  men  in  1  day  earn  $10|,  in 
9  days  they  will  earn  9  times  $10i,  or  $94^,  the  same  as  before  found. 

Third  Explanation  {tnore  abbreviated). — If  5  men  in  4  days,  doing  20  days'  work,  earn 
$30,  $11  would  equal  1  daj''s  work  ;  7  men  in  9  days  do  63  days'  work,  and  since  1  day's  work 
equals  $1|,  63  days'  work  will  equal  $941,  as  before  found. 

Example  3.  If  6  men  can  cut  45  cords  of  wood  iu  3  days,  how  many  chords 
can  8  men  cut  in  9  days  ? 

First  Explanation  (extended). — If  6  men  cut  45  cd.  in  3  days,  in  1  day,  which  is  J  of  3 
days,  they  can  cut  ^  of  45  cd.,  or  15  cd.;  and  if  6  men  can  in  1  day  cut  15  cd.,  1  man  in  1  day 
can  cut  J  of  15  cd.,  or  21  cd. ;  since  1  man  in  1  day  can  cut  21  cd.,  8  men  can  in  1  daj'  cut  8 
times  2^  cd.,  or  20  cd.;  and  if  8  men  in  1  day  can  cut  20  cd.,  in  9  days  they  can  cut  180  cd. 

Second  Explanation  (abbreviated). — 6  men  in  3  days,  doing  18  days'  work,  cut  45  cd.; 
hence  21  cd.  can  be  cut  by  1  man  in  1  day ;  then  8  men  in  9  days,  doing  72  days'  work,  can 
cut  72  times  2i  cd.,  or  180  cd.,  as  before  found. 

Example  J.  If  a  post  4  ft.  high  casts  a  shadow  13  ft.  in  length,  wliat  must 
be  the  height  of  a  post  that  Avill  cast  a  shadow  125  ft.  in  length? 

Explanation. — If  a  post  4  ft.  high  casts  a  shadow  13  ft.,  a  post  1  ft.  high  would  cast  a 
shadow  3]  ft. ;  since  a  shadow  3]  ft.  is  cast  by  a  post  1  ft.  high,  a  post  that  will  cast  a  shadow 
125  ft.  in  length  must  be  as  many  times  1  ft.  in  height  as  3]  ft.  are  contained  times  in  125  ft., 
or  SSx'V  ft. 


Example  4-     If  the  hour  and  minute  hands  of  a  clock  are  together  at  noon, 
;  what  times  af 
4  and  5  o'clock  ? 


at  what  times  after  noon  will  they  again  be  together  ?    At  what  time  between 


88  ANALYSIS. 

Explanation. — Since  the  minute  hand  passes  the  hour  hand  11  times  in  12  hours,  it  will 
pass  it  the  first  time  in  i\  of  12  hours;  the  second  time  in  f^  of  12  hours;  the  third  time  in  y\ 
of  12  hours;  the  fourth  time  in  y\  of  12  hours.  y\  of  12  hours  equals  4  hours,  21  minutes, 
and  49^^f  seconds  ;  therefore  the  hands  will  he  together  between  4  and  5  o'clock  at  21  minutes 
4.9^j  seconds  after  4  o'clock. 

Remark. — Apply  the  same  reasoning  to  all  examples  of  this  class. 

Example  o.  If  Grace  were  ^  older  than  she  is,  her  age  would  equal  ^  of  her 
grandmother's.     What  is  the  age  of  each,  if  the  age  of  both  is  87  years  ? 

Explanation. — If  Grace  were  l  older  than  she  is,  she  would  be  |  of  her  present  age  ;  and 
since  if  she  were  !;  her  present  age,  she  would  be  only  J  as  old  as  her  grandmother,  the  age  of 
grandmother  must  be  4  times  ?  or  V  of  the  age  of  Grace,  and  the  age  of  both  must  be  |  +  V" 
or  -J*  of  the  age  of  Grace  ;  since  the  age  of  both  is  87  years,  87  years  must  be  -\"-  of  the  age  of 
Grace,  who  must  be  15  years  old.  If  Grace's  age  be  increased  by  ^  of  itself,  or  3  years,  she 
will  be  18  years  of  age ;  and  since  her  age  would  then  be  only  J  of  grandmother's  age,  the 
age  of  grandmother  must  be  4  times  18  years,  or  72  years. 

Example  6.  A  mau  being  asked  his  age,  replied  :  "My  father  was  born  in 
1805  and  my  mother  in  1806  ;  the  sum  of  their  ages  at  the  time  of  my  birth  was 
two  and  one-third  times  my  age  in  1851."     How  old  was  the  man  in  1888  ? 

Explanation, — If  the  father  was  born  in  1805  and  the  mother  in  1806,  the  sum  of  their 
ages  in  1851  was  91  years  ;  and  since  the  sum  of  their  ages  at  the  time  of  the  birth  of  the  son 
was  2^  times  his  age  in  1851,  and  the  parents  each  increased  in  years  after  the  son's  birth  as 
fast  as  he  did,  in  1851  the  sum  of  their  ages  must  have  been  4|^  times  the  age  of  the  son;  hence 
the  son,  in  1851,  was  91  years  -i-  4^,  or  21  years  of  age,  and  he  must  have  been  born  in  1830,  and 
in  1888  would  be  58  years  old. 

7.  The  sum  of  two  numbers  is  65,  and  their  difference  is  equal  to  ^  of  the 
greater  number.     Find  the  two  numbers. 

8.  How  long  after  noon  will  it  be  when  the  minute  hand  passes  the  hour 
hand  the  third  time? 

9.  How  long  after  noon  will  it  be  when  the  minute  hand  joasses  the  hour 
hand  the  eleventh  time? 

10.  A's  age  is  2|  times  the  age  of  B,  and  the  age  of  C  is  2^^  times  the  age  of 
both  A  and  B.     If  the  sum  of  their  ages  is  116  years,  what  is  the  age  of  each? 

11.  A  man  bought  15  bushels  of  barley,  and  36  bushels  of  oats,  for  $38.80,  and 
25  bushels  of  barley,  18  bushels  of  oats,  for  $29.10.  How  much  per  bushel  did 
he  give  for  each  kind  of  grain? 

12.  Charles,  when  asked  his  age,  replied:  "  My  father  was  born  in  1843,  and 
my  mother  in  1847.  The  sum  of  their  ages  at  the  time  of  my  birth  was  5  times, 
my  age  in  1887."     In  what  year  will  Charles  be  25  years  of  age  ? 


SPECIAL   APPLICATIONS.  8& 


SPECIAL    APPLICATIONS. 

295.  Special  Applications,  as  here  treated,  embraces  the  use,  in  the 
sohition  of  problems,  of  any  or  all  explanations  heretofore  given,  and  the  con- 
sideration of  cost,  price,  and  quantity y  as  being  the  elements  of  every  business 
transaction  ;  it  also  treats  of  such  contracted  methods  as  may  be  employed  in 
dealing  with  aliquot  parts  of  the  powers  of  10,  or  of  other  numbers. 

General  Rules. — 1.  If  the  price  and  quantity  he  given,  the  cost  may 
he  found  hy  multiplying,  the  price  hy  the  quantity. 

2.  If  the  cost  and  quantity  he  given,  the  price  may  he  found  hy  divid- 
the  cost  hy  the  quantity. 

3.  If  the  cost  and  j/rice  he  given,  the  quantity  may  he  found  hy  divid- 
ing the  cost  hy  the  price. 

ALIQUOT    PARTS. 

296.  The  Aliquot  Parts  of  a  number  are  the  even  parts  of  that  number. 
25,  33i,  \%\,  are  aliquot,  or  even,  parts  of  100, 

Remakk — The  component  factors  of  a  number  must  be  integral,  while  the  aliquot  parts  of  a 
number  may  be  either  integral  or  mixed. 

297.  The  even  parts  of  other  even  parts  may  be  called  parts  of  parts  ;  as, 
i  =  i  of  i  ;  or,  since  33|  is  a  part  of  100,  ^  of  33^,  or  11^,  must  be  a  part  of  the 
part  33^. 

Remakk — Full  illustrations  of  the  use  of  aliquot  parts  will  follow.  Those  of  $1,  equal  to 
100^,  being  the  most  valuable  for  use,  will  be  mainly  considered. 


1.  50  cents  =  ^  of  $1 

3.  33^  cents  =  ^  of  |] 

3.  25  cents  =  -]-  of  |1 

4.  20  cents  = -\  oi  U. 


Aliquot  Parts  of  One  Dollar. 

5.     16f  cents  =  i  of  $1. 


10  cents  =  J„  of  |1. 
8^  cents  =  ^  of  $1. 


9.  6^  cents  =  iV  of  $1. 

10.  3|  cents  =  ^  of  $1. 

11.  2i  cents  =  ^V  of  U. 

12.  If  cents  =  ^^  of  $1. 


Aliquot  Parts  of  Aliquot  Parts  of  One  Dollar. 

<o\  cents  =  ^  of  25  cents.    I  5  cents  =  -^  of  50  cents,  i    \%\  cents  =  \  of  50  cents. 
124  cents  =  -|-  of  25  cents.  I  6^  cents  =  ^  of  50  cents.   |    25  cents  =  -i  of  50  cents. 

Suggestion  to  Teachek. — Let  each  one  of  the  following  conditions  be  given  to  the  class 
as  a  question,  the  required  answer  to  which  is  the  rule. 


90  ALIQUOT   PARTS. 

INSTRUCTIONS  FOR  PRACTICE  WITH    ALIQUOT    PARTS. 

298.     1.     To  find  tlie  cost  of  a  quantity  when  tlio  price  of  1  is  50   cents. 
KULE.  —  Consider  the  quant  it  if  as  ilallors,  and  divide  In/  ^. 

2.  To  find  the  cost  when  tlie  prirp  of  1  is  33^{i*.      Eule. — Divide  the  quantity, 
considered  as  dollars,  by  S. 

3.  To  find  the  cost  when  i\\e price  of  1  is  25^^.     Elle. — Divide  the  quantity, 
considered  as  dollars,  ly  Jf. 

4.  To   find   the  cost  at   20'/.     Rule. — Divide   the    quantity,    considered  as 
dollars,  by  5. 

5.  To  find  the  cost  at  16f '/.    Kile. — Divide  the  quantity,  considered  as  dollars, 
by  6. 

6.  To  find  tlie  cost  at  12+^'.    Rule. — Divide  the  quantity,  considered  as  dollars, 
by  S. 

7.  To  find  tliecos^at  8^^.    Rule. — Divide  the  quantity,  considered  as  dollars, 
by  12 

S.     To  find  the  cost  at  6^^.    Rule. — Divide  the  quantity,  considered  as  dollars, 
by  JO. 

9.  To  find  the  cost  at  10^.     Rule. — Point  off  from  the  right  one  place  in 
the  quantity,  and  consider  as  dollars. 

10.  To  find  the  cost  at  5^^      Rule. — Point  off  one  place  in  the  quantity, 
consider  as  dollars,  and  divide  by  2. 

11.  To  find  the  cost  at  3^f.     Rule. — Point  off  one  place  in  the  quantity, 
consider  as  dollars,  and  divide  by  3. 

12.  To  find  the  cost  at  2^^.      Rule. — Point  off'  one  place  in  the  quantity, 
consider  as  dollars,  and  divide  by  Jf. 

13.  To  find  the  cost  at  1|^/.      Rule. — Point  off'  one  place  in  the  quantity, 
consider  as  dollars,  and  divide  by  6. 

IJf..     To  find  the  cost  at  1^^.     Rule. — Point  off  one  place  in  the  quantity, 
consider  as  dollars,  and  divide  by  8. 


MISCELLANEOUS   CONTRACTIONS. 

299.     1.     To  find  tlie  cost  when  x.\\q  price  of  1  is  75  cents.     Rule. — From  the 
quantity,  considered  as  dollars,  take  ^  of  itself. 

2.  To  find  the  cost  when  the  price  of  one  is  80^.     Rule. — From  the  quantity, 
considered  as  dollars,  take  I  of  itself. 

3.  To  find  tlie  cost  when  the  i)rice  of  one  is  66f  ^.     Rule. — From  the  quantity, 
considered  as  dollars,  take  \  of  itself. 

Jf.     To  find  the  cost  when  the  price  of  one  is  ^1.25.  Rule. — To  the  quantity, 
considered  as  dollars,  add  \  of  itself. 

5.  To  find  the  cost  when  the  price  of  one  is  11.50.  Rule. — To  the  quantity, 
considered  as  dollars,  add  \  of  itself.  v 

6.  To  find^he  cost  when  the  price  of  one  is  12.50.  Rule. — Annex  a  cipher 
to  the  quantity,  consider  as  dollars,  and  divide  by  Jf. 


INSTRUCTIONS    FOR    FINDING    QUANTITY.  91 

7.  To  find  the  cost  when  the  price  of  one  is  $7.50.  Umlv..— Annex  a  cipher 
to  the  quantity,  consider  as  dollars,  and  subtract  \. 

8.  To  find  the  cost  when  the  price  of  one  is  G|^-.  Rule. — Point  off  one  place 
in  the  quantity,  consider  as  dollars,  and  subtract  \. 

9:  To  find  the  cost  Avhen  the  price  of  one  is  13^'/.  Rule. — Point  off  one  place 
in  the  quantity,  co7isider  as  dollars,  and  add  ^. 

10.  To  find  the  cost  wlien  the  price  of  one  is  $1.33^.  Rule. — Add  I  to  the 
quantity,  and  consider  as  dollars. 

11.  To  find  the  cost  when  the  jn-ice  of  one  is  §1.10.  Rule. — Add  -^V  to  the 
quantity,  and  consider  as  dollars. 

12.  To  find  the  cost  when  the  ^v\ce.  of  one  is  $1.20.  Rule. — Add\  to  the 
quantity,  and  consider  as  dollars. 

13.  To  find  the  cost  when  the  price  of  one  is  $1.35.  Rule. — Add  \  and  -^ 
to  the  quantity,  and  consider  as  dollars. 

lU.  To  find  the  cost  when  the  price  of  one  is  11.75.  Rule. — Add  \  and  \  to 
the  quantity,  and  consider  as  dollars. 

15.  To  find  the  cost  when  tlie  price  of  one  is  $3.33^.  Rule. — Annex  a  cipher 
to  the  quantity,  consider  as  dollars,  and  divide  by  3. 


INSTRUCTIONS   FOR   FINDING   QUANTITY. 

300.  1.  To  find  the  quantity  of  articles  tliat  any  given  sum  of  money  will 
purchase,  when  the  price  of  one  is  50^'.  Rule. — Multiply  the  dollars,  considered 
as  quantity,  by  2. 

2.  To  find  the  quantity  when  the  price  of  one  is  33-^^'.  Rule. — Multiply  the 
dollars,  considered  as  quantity,  by  3. 

3.  To  find  the  quantity  when  the  price  of  one  is  25^-.  Rule. — Multiply  the 
dollars,  considered  as  quantity,  by  Jf. 

Jf..  To  find  the  quantity  when  the  price  of  one  is  20^-.  Rule. — Multiply  the 
dollars,  considered  as  quantity,  by  5. 

6.  To  find  the  quantity  when  the  price  of  one  is  16|^'.  Rule. — Multiply  the 
dollars,  considered  as  quantity,  by  6. 

6.  To  find  the  quantity  when  the  i)rice  of  one  is  1249^-.  Rule. — Midtiply  the 
dollars,  considered  as  quantity,  by  8. 

7.  To  find  the  quantity  when  the  i)rice  of  one  is  10^'.  Rule. — Annex  a  cipher 
to  the  dollars,  and  consider  as  quantity. 

8.  To  find  the  quantity  when  the  price  of  one  is  8^^-.  Rule. — Multiply  the 
dollars,  considered  as  quantify,  by  12. 

0.  To  find  the  quantity  when  the  price  of  one  is  6^^'.  Rule. — Multiply  the 
dollars,  considered  as  quantity,  by  16 

10.  To  find  the  quantity  when  the  price  of  one  is  5^^.  Rule. — Annex  a  cipher 
to  the  dollars,  consider  as  quantity,  and  multiply  by  2. 

11.  To  find  the  quantity  when  the  price  of  one  is  3^^.  Rule. — Annex  a 
cipher  to  the  dollars,  consider  as  quantity,  and  multiply  by  3. 


92  MISCKLLAXEOUS   COXTRACTIONS. 

12.  To  find  the  quantity  when  the  price  of  one  is  'l^<p.  Rule. — Ayinex  a 
cipher  to  the  dollars,  consider  as  qiiantity,  and  multiply  by  4. 

IS.  To  find  the  quantity  when  the  price  of  one  is  \\f.  Rule. — Annex  a 
cipher  to  the  dollars,  consider  as  quantity,  and  multiply  by  6. 

14.  To  find  the  quantity  when  the  price  of  one  is  1\</:  Rule. — Annex  a 
cipher  to  the  dollars,  consider  as  quantity,  and  multiply  by  8. 

MISCELLANEOUS   CONTRACTIONS. 

301.  1.  To  find  the  quantity  when  the  price  of  one  is  %\.'lb.  Rule. — 
Point  off  one  place  in  the  dollars,  consider  as  quantity,  and  multiply  by  8. 

2.  To  find  the  quantity  when  the  price  of  one  is  $1.66f.  Rule. — Point  off 
one  place  in  the  dollars,  consider  as  quantity,  and  multiply  by  6. 

3.  To  find  the  quantity  when  the  price  of  one  is  $2.50.  Rule. — Point  off 
one  place  in  the  dollars,  consider  as  quantity,  and  multipty  by  4- 

Jf..  To  find  the  quantity  when  the  price  of  one  is  86.66|.  Rule. — Point  off 
one  j)lace  in  the  dollars,  consider  as  quantity,  and  add  \. 

5.  To  find  tlie  quantity  when  the  price  of  one  is  §7.50.  Rule. — Point  off 
one  place  in  the  dollars,  consider  as  quantity,  and  add  ^. 

6.  To  find  the  quantity  when  the  price  of  one  is  S12.50.  Rule. — Poitit  off 
two  places  in  the  dollars,  consider  as  qua)itity,  and  multiply  by  8. 

MISCELLAXEOUS   EXAMPLES   IX    FrXDIXG   QUANTITY. 

302.  Example  i. — How  many  pounds  of-  tea,  worth  66|^'  per  lb.,  can  be 
bought  for  1147  ? 

Explanation. — Since  the  price  of  one  pound  is  contained  1^  times  in  $1,  the  number  of 
pounds  bouo:ht  will  be  1^  times  the  number  of  dollars  invested  ;  hence,  add  to  the  number  of 
dollars  (as  pounds)  |  of  itself,  and  the  result  will  be  the  number  of  pounds  purchased. 

Example  ~. — How  many  pounds  of  tea,  at  T5^  per  lb.,  will  8419.25  purchase  ? 

Explanation. — Since  the  price  of  one  pound  is  contained  1^  times  in  $1,  the  number  of 
pounds  bought  will  be  1^  times  the  number  of  dollars  invested  ;  hence,  add  to  the  number  of 
dollars  ^  of  itself,  and  the  result  will  represent  the  number  of  pounds  purchased. 

Example  3. — How  many  pounds  of  tea,  at  S7^<p  per  lb.,  can  be  bought  for 
$316  ? 

Explanation.  —Since  the  cost  of  one  pound  is  contained  li  times  in  $1,  the  number  of 
pounds  purchased  will  be  |  greater  than  the  dollars  invested. 

Ex.\MPLE  4- — At  83^^  ]ter  yard,  how  many  yards  of  cloth  can  be  bought  for 
$1128.50? 

Explanation. — Since  the  price  of  one  yard  is  contained  IJ  times  in  $1,  we  can  buy  | 
more  yards  than  we  have  dollars  to  invest. 

Example  o. — At  80^*  per  yard,  how  many  yards  can  be  bought  for  $246.25  ? 

Explanation. — Since  $1  will  buy  1 J  yards,  $246.25  will  buy  246.25  times  1\  yards  ;  hence, 
to  the  number  of  dollars  add  \  of  itself. 

Remark. — The  teacher  can  profitably  extend  these  exercises  for  mental  and  written  drill 
for  pupils. 


MISCELLANEOUS   EXAMPLES    IN    FINDING    QUANTITY.  93 

303.     To  Find  the  Cost  when  the  Price  is  an  Aliquot  Part  of  a  Dollar. 
Example. — Required,  the  cost  of  546  gallons  of  molasses,  at  33^^  per  gallon  ? 

Explanation.— Since  33y  is  ^  of  $1,  3  gallons  would  cost  $1 ;  and  if  $1  -will  buy  3  gallons, 
it  will  require  as  many  dollars  to  buy  546  gallons  as  3  is  contained  times  in  546,  or  $182. 

Rule. —  Divide  the  quantity,  considered  as  dollars,  by  the  number  of 
units  of  the  quantity  that  ivill  cost  $1. 

EXAMPLES   FOK   PRACTICE. 


304.     Find  the  cost  of 

1.  286  lb.  of  tea,  at  50^  per  lb. 

2.  1152  yd.  of  linen,  at  33^^  per  yd. 
•5.  527  lb.  of  lard,  at  124^  per  lb. 

4.  455  gal.  of  molasses,  at  25^-  per  gal. 

5.  1751  doz.  of  eggs,  at  16f^  per  doz. 

6.  2133  lb.  of  pork,  at  ^\<fi  per  lb. 

Remark. — In  the  following  examples  treat  the  amount  of  each  item  as  a  separate  result, 
and  consider  5  or  more  mills  as  1^. 


1238  cans  of  salmon,  at  20^  per  can. 
756|  gal.  of  cider,  at  8^^  per  gal. 
81  lb.  of  meal,  at  \\<f:  per  lb. 
Ill  qt.  of  berries,  at  ?>\<P  per  qt. 

11.  1354^  yd.  of  cotton,  at  5^'  per  yd. 

12.  840  lb.  of  salt,  at  \\<p  per  lb. 


9. 
10. 


13. 


Find  the  total  cost  of  the  following  : 


86  yd.,  at  12|^  per  yd. 
93  yd.,  at  ^(f;  per  yd. 
150  yd.,  at  25^  per  yd. 


591  yd.,  at  10^  per  yd. 
327.yd.,at33^^peryd. 
1141  yd.,  at  25^  per  yd. 


1600  yd.,  at  16f^  per  yd. 
71  yd.,  at  bQ<ji  per  yd. 
947  yd.,  at  3|^  per  yd. 


Remark. — Quarters  are  often  written  thus:  5*  —  5f ;  17^  =  17^  ;  IP  =  11| ;  this  method 
is  not  used  with  other  fractions. 


IJf.     Find  the  total  cost  of  the  following  : 
832  ya.,  at  6i^  per  yd.        1272  yd.,  at  16|^  per  yd. 


713  yd.^  at  8^^  per  yd. 
230^  yd.,  at  25^  per  yd. 


1000  yd.,  at  If {Zi  per  yd. 
547*  yd.,  at  b<^  per  yd. 


2855^  yd.,  at  20^  per  yd. 
8722  yd.,  at  50^  per  yd. 
624    yd.,  at  2i^  per  yd. 


305.  To  Find  the  Cost  when  the  Price  is  given  and  the  Quantity  is  a 
Multiple,  or  an  Aliquot  part,  of  100  or  1000. 

Remark.— When  the  quantity  is  in  even  hundreds  or  thousands,  find  the  cost  by  multiply- 
ing the  price,  expressed  as  a  decimal,  by  the  number  of  hundreds  or  thousands.  For  parts  of 
hundreds  or  thousands,  add  equivalent  fractional  parts. 

Example  7.— Find  the  cost  of  100  yards  of  cloth,  at  54^^  per  yard. 

Explanation.— Since  1  yard  costs  %  .5425, 100  yards  will  cost  100  times  as  much,  or  $54.25. 

Example  i?.— Find  the  cost  of  300  yards,  at  17^^  per  yard. 

Explanation.— Since  1  yard  costs  $  .175,  100  yards  will  cost  $17.50,  and  300  yards,  or  3 
times  100  yards,  will  cost  3  times  $17.50,  or  $52.50. 

Example  5.— Find  the  cost  of  1000  yards,  at  83J^  per  yard. 
Explanation.— Since  1  yd.  costs  $.8375, 1000  yd.  will  cost  1000  times  $  .8375,  or  $837.50. 


94  EXAMPLES    FOR    PRACTICE   IN    SPECIAL    A PPLICATIOIf S. 

Example  4. — Find  the  cost  of  75  yards,  at  81.37^  per  yard. 
Expi>AKATTOx.— Since  1  yard  costs  f  1.37i,  100  yards  will  cost  $137.50  ;  7.5  yanls  will  cost 
1  less  than  $137.50,  or  $103,125,  or  $103.13. 

Example  .7. — Find  the  cost  of  250  yards,  at  $1.75  per  yard. 
ExPT.A^^\TIo^•.— Since  1  yard  costs  $1.75,  1000  yards  will  cost  $1750  :   250  yards,  or  }  of 
1000  yards,  will  cost  \  of  $1750,  or  $437.50. 

Example  6.     Parts  of  Parts. — What  will  be  the  cost  of  1420  bushels  of 
wheat,  at  $1.37^  per  bushel  ? 

ExPLANATiox.— At  $1  per  bushel.  1426  bushels  will  cost $1426.00 

At  \  =  25o  per  bushel,  1426  bushels  wUl  cost 356.50 

At  I  =  (i  of  j),  or  12io  per  bushel,  1426  bushels  will  cost        178.25 

At  $1,371  per  bushel,  1426  bushels  will  cost $1960.75 

Example  7. — What  is  the  cost  of  824  yards  of  cloth,  at  $1.75  per  yard  ? 

ExPLAXATiox.— At  $1  per  yard,  824  yards  will  cost $824.00 

At    1  =  50o  per  yard,  824  yards  will  cost 412.00 

At    i  =  250  i>er  yard,  824  yards  will  cost 206.00 

At  $1.75  per  yard,  824  yards  will  cost $1442.00 

Example  S. — At  55^  per  lb.,  what  will  l)e  the  cost  of  14G  lb.  of  gunpowder  ? 
ExPLA^-ATI0^-.— At  $1  per  lb.,  146  1b.  will  cost $146.00 

At  ^  =  50C  per  lb.,  146  1b.  will  cost.. 73.00 

At  ^  =  (iV  of  i),  or  5/;*  per  lb. .  146  lb.  will  cost         7.30 

AtSS^  per  lb.,  146  lb.  will  cost $80.30 


KXAMPLES   FOR  WRITTKN    PKACTICK. 


306.  Find  the  cost  of 
2U0  lb.,  @  37^^.  i 
700  lb.,  @  51i^. 
150  lb.,  @  14f^. 
250  lb.,  @  21f^. 
1000  lb.,  @  $1.12^. 
750  lb.  @  81f'/. 
1250  lb.,  @  $2.62^. 


8.  400  lb.,  @.  95i^.  lo. 

9.  250  lb.,  @  9f^.  1  16. 

10.  75  lb.,  @  60J^-.  '  17. 

11.  125  lb.,  @  27^'/.  j  18. 

12.  1100  1b.,  @  $1,424.  I  19. 

13.  500  lb.,  @.  374^.  j  20. 
U.  1500  1b.,  @  18f^.  '  21. 


300  lb.,  @41|^. 
3000  lb.,  @  12^^. 
2500  Ik,  @  61.10. 
25  lb.,  @  $1.85. 
150  lb.,  @  334^. 
75  lb.,  @,  $1.15. 
125  lb.,@  $1.25. 


KX.VMPLKS   KOK   MKNTAL   PKACTICK. 


Remark. — All  extensions  in  the  following  examples  should  be  made  mentally,  the  pupil 
writing  only  the  cost  of  each  item  for  footing. 


307.     1.     Find  the  total  cost  of 


516  lb.,  at  \0<f:  per  lb. 

484  lb.,  at  5^  per  lb. 
1000  lb.,  at  74^!'  per  lb. 
2500  1b.,  at  8^  per  lb. 
3000  1b.,  at  11^  per  lb. 


21 G  lb.,  at  124^  per  lb. 
1120  lb.,  at  50^  per  lb. 

818  lb.,  at  25^  per  lb. 
1400  lb.,  at  20^  per  lb. 

381  lb.,  at  40^  per  lb. 


1095  lb.,  at  33^^  per  lb. 

125  lb.,  at  6^*  per  lb. 

711  lb.,  at  30^  per  lb. 

97  lb.,  at  ^<f■  i)er  lb. 

150  1b.,  at  Gi^per  lb. 


EXAMPLES    FOR    MKNTAL    PRACTICE. 


95 


,?.     Find  the  total  cost  of 

G86  yd.,  at  15^/  per  yd.  297  yd.,  at  25^'  per  yd. 

2140  yd.,  at  5^  per  yd.  1100  yd.,  at  439^-  per  yd. 

853  yd.,  at  10^  per  yd.  1200  yd.,  at  28^^'  per  yd. 

246  yd.,  at  20^/  per  yd.  298  yd.,  at  50^  per  yd. 

398  yd.,  at  30^'  per  yd.  931  yd.,  at  25^  per  yd. 

450  yd.,  at  33^^  per  yd.  1315  yd.,  at  33^^  per  yd. 

3.     Find  the  total  cost  of 

1400  lb.,  at  4^  per  lb.  93G2  lb.,  at  12.}^'  per  lb. 

2168  lb.,  at  3^^  per  lb.  2143^  lb.,  at  15^  per  lb. 

7000  lb.,  at  bi  per  lb.  540  lb.,  at  11^/  per  lb. 

2462  lb.,  at  Gy  per  lb.  2980  lb.,  at  lG|r/  j.cr  lb. 

5963  lb,,  at  8^^  per  lb.  593  lb.,  at  13^  per  lb. 

12521  lb.,  at  10^  per  lb.  12503  ib.,  at  6^^-  per  lb. 

i.     Find  the  total  cost  of 


5252  yd.,  at  8^"  per  yd. 
11781  yd.,  at  9^  per  yd. 
28533  yd.,  at  10^  per  yd. 
1400  yd.,  at  G4j^  iwv  yd. 

■'.     Find  the  total  cost 

832  yj,^  at  55^  per  yd. 

713  yd.,  at  75^  per  yd. 

1071  yd.,  at  bOf  per  yd. 

2303yd.,atG6|^peryd. 

17532  yd.,at25{?5peryd. 

46  yd.,  at  15^  per  yd. 

6.     Find  the  total  cost 
629*  yd.,  at  3^^'  per  yd. 

11402  yd.,  at  5^  per  yd. 
5943  yd.,  at  6^^  per  yd. 

34G9yd.,  at  8J^  per  yd. 

12912  yd.,  at  11^  per  yd. 
5933  yd.,  at  12^^^  per  yd. 


367  yd.,  at  81.25  per  yd. 
282  yd.,  at$2.50peryd. 
5771  yd.,  at  55^  per  yd. 
315  yd,,  at  75^  i)er  yd. 


of 


of 


1272  yd.,  at  16|^  per  yd. 

500  yd., -at  18|^/  per  yd. 

2G93  yd.,  at  124^/ per  yd. 
29601  yd.,  at  9o'^  i)er  yd. 

1832  yd.,  at  8^^  per  yd. 
23753  yd.,  at  10^'  per  yd. 

250  yd.,  at  13g^  per  yd. 

400  yd.,  at  15^^  per  yd. 

7563  yd.,  at  16|^  per  yd. 
13751  yd.,at  20^per  yd. 
1741  yd.,  at  259^' per  yd. 

9063  yd.,  at  66f^  per  yd. 


800  yd.,  at  139^- per  yd. 

959  yd.,  at  16:^9^-  per  yd. 
1000  yd.,  at  19f^-peryd. 
2000  yd.,  at  21§^  per  yd. 

606  yd.,  at  124^- per  yd. 

150  yd.,  ar  25^/  per  yd. 

291  lb.,  at  50^- per  lb. 
14372  lb.,  at  250  per  lb. 
19783  lb.,  at  33^0  per  lb. 

8441  ib.^  at  750  per  lb. 

930  1b.,  at  66|0perlb. 

6752  lb.,  at  1240  per  lb. 

2100  yd.,  at  750  per  yd. 

146  yd.,  at  250  per  yd. 

500  yd.,  at  8. 3  7^  per  yd. 
1000  yd.,  at  81. 87^  per  yd 

20053  yd.,  at  6i0  peryd. 
1000  yd.,  at  83f  0'per  yd. 

250  yd.,  at  27^0  per  yd. 

9312  yd.,  at  33^0  per  yd. 

7683yd.,at$  1.25  per  yd. 
17561yd., at  81.124  peryd. 

55482  yd.,  at  140  per  yd. 
1250  yd.,  at  740  per  yd. 

300  yd.,  at  2310  per  yd. 

500  yd.,  at  41^0  per  yd. 

186  yd.,  at  160  per  yd. 

175  yd.,  at  150  per  yd. 


308.     To  Find  the  Quantity  when  the  Price  is  an  Aliquot  Part  of  $1. 

Example  /. — If  oats  cost  33^0  per  bushel,  how  many  bushels  can  ])e  bought 
for  $54  ? 

Explanation. —Since  1  bushel  costs  33JY,  or  \  of  $1,  3  bushels  can  be  bought  for  $1  ;  and 
if  |1  will  buy  3  bushels,  $54  will  buy  54  times  3  bushels,  or  162  bushels. 

Example  ~. — If  a  yard  of  cloth  costs  66|0,  how  numy  yard.s  will  >5,S4  l)uy  ? 
Explanation. — Since  the  price  is  \  of  itself  less  than  $1  per  yard,  the  number  of  yards 
willbei  greater  than  the  number  of  dollars  expended;  .\  of  84  =42;  84  +  42  —  126,  or  126  yards. 


96  EXAMPLES   FOR   PRACTICE. 

Example  3. — At  87^^  per  bushel,  how  many  bushels  of  wheat  can  be  bought 
for  *12G:  ? 

ExTLAJNATiON-.— Since  the  price  is  i  of  itself  less  than  $1  per  bushel,  the  number  of  bushels 
will  be  i  greater  than  the  number  of  dollars  expended;  i  of  1267  =  181;  181  -f- 1267  =  1448, 
or  1448  bushels. 

Remark. — Application  of  the  principle  of  reciprocals  can  profitably  be  introduced  at  this 
point;  the  reasoning  will  be  the  same  as  in  the  examples  given  above. 

Example  4. — At  66|^  per  yard,  how  many  yards  of  cloth  can  be  bought  for  $84? 
Explanation. — 66|^  =  $|;  write  its  reciprocal,  |,  and  multiply  by  $84. 

Example  o. — At  75^  per  yard,  liow  many  yards  of  cloth  can  be  bought  for  $84? 
Explanation. — 75^  =  $J;  write  its  reciprocal,  |,  and  multiply  by  $84. 

Example  6. — At  87^^  per  yard,  how  many  yards  of  cloth  can  be  bought  for  $84? 
Explanation. — 81i^  —  $i;  write  its  reciprocal,  i.  and  multiply  by  $84. 

Rules. — 1.  Multiply  tJie  cost  hij  the  quantity  that  can  he  bought 
for  $1.    Or, 

2.  Add  to  the  cost  (as  qioantity)  such  a  paH  of  itself  as  the  price 
lacks  of  being  $1. 

EXAMPLES  FOK   PRACTICE. 

309.  1.  If  1  lb.  of  candy  can  be  bought  for  25^,  how  many  pounds  can  be 
bought  for  $5.75  ? 

~.     At  33^^  per  yard,  how  many  yards  of  cloth  will  $1542.50  buy  ? 

3.  A  boy  expended  $1  for  almonds,  at  16§^  per  lb.  How  many  pounds  did 
he  buy  ? 

-4.     At  75^  per  yard,  how  many  yards  of  cloth  can  be  bought  for  $572.40  ? 

5.  If  I  invest  $175.30  in  eggs,  at  20^  per  doz.,  how  many  dozens  do  I  purchase? 

6.  A  farmer  sold  26^^  bu.  buckwheat,  at  87^^  per  bu.,  and  took  his  pay  in  sugar 
at  6^^  per  lb.     How  many  pounds  should  he  have  received  ? 

7.  A  gardener  exchanged  132  qt.  of  berries,  at  8^^  per  qt.,  and  75  doz.  corn,  at 
12^^  per  doz.,  for  cloth  at  25^  per  yd.     How  many  yards  did  he  receive  ? 

8.  If  I  exchange  1920  acres  of  wild  land,  at  $7.50  per  acre,  for  an  improved 
farm  at  $125  per  acre,  what  should  be  the  number  of  acres  in  my  farm  ? 

9.  A  farmer  gave  8 J  cwt.  of  pork,  at  $7.50  per  cwt.,  15  bu.  of  beans,  at  $3.25 
per  bu.,  and  4Gi-  bu.  of  oats,  at  33^^  per  bu.,  for  28  yd.  of  dress  silk,  at  $1.25  per 
yd.,  and  52^  yd.  of  delaine,  at  16|^  per  yd.,  receiving  for  the  remainder,  cotton 
goods  at  12^^  per  yd.  How  many  yards  of  cotton  goods  should  be  delivered  to 
him  ? 

10.  "When  potatoes  are  worth  G6|^-  per  bu.,  and  turnips  25^  per  bu.,  how  many 
pounds  of  coffee,  at  16|5#  per  lb.,  will  2)ay  for  24  bu.  of  potatoes  and  18  bu.  of 
turnips  ? 

11.  Having  bought  1487  lb.  A.  sugar,  at  6^^  per  lb.;  872  lb.  C.  sugar,  at  5^ 
per  lb. ;  628^  lb.  Y.  H.  tea,  at  33^^  per  lb. ;  522  lb.  J.  tea,  at  25/  per  lb. ;  650  lb. 
Rio  coffee,  at  12^/  per  lb.;  and  81  sacks  of  flour,  at  $1.25  persadk,  I  give  in  pay- 
ment seven  one-hundred  dollar  bills.     How  much  should  be  returned  to  me? 


EXAMPLES   FOR    PKACTICE.  97 

310.  To  find  the  Cost  of  Articles  Sold  by  the  C. 
C  stands  for  100.     M  stands  for  1000. 

Example. — What  is  the  cost  of  416  lb.  phosphate,  at  $2.00  per  hundred? 
ExPLAKATiON.— 416  Ibs.  =  4.16  hundred  lbs.     If  1  hundred  pounds  cost  $2.00,  4.16  hundred 
lb.  will  cost  4.16  times  $2,  or  $8.32. 

Rule. — Reduce  the  quantity  to  hundreds  and  decimals  of  a  hundred, 
by  pointing  off  two  places  from  the  right,  then  multiply  hy  the  pi'ice  per  C. 

EXAMPLES   FOR   PKACTICE. 

311.  Find  the  cost  of 


1.  1753  lb.  of  salt,  at  $1.25  per  C. 

£.  8425  lb.  of  scrap  iron,  at  $1. 10  per  C. 

3.  2156  lb.  of  fence  wire,  at  $3.25  per  C. 

4,  378  fence  posts,  at  $7. 50  per  C. 

■5.  3295  lb.  of  gitano,  at  $4.50  per  C. 


6.  905  lb.  of  lead,  at  $3.50  per  C. 

7.  1125  lb.  of  castings,  at  $2.25  per  C. 

8.  1620  handles,  at  $5.50  per  C. 

9.  509  lb.  of  beef,  at  $12.50  per  C. 
10.  23765  lb.  of  nails,  at  15^  per  C. 


312.  To  Find  the  Cost  of  Articles  Sold  by  the  M. 
Example. — At  $7.00  per  M,  what  will  be  the  cost  of  1544  bricks? 

ExPLA^fATI0N. — 1544  bricks  =  1.544  thousand  bricks;  and  if  one  thousand  bricks  coat  $7, 
1.544  thousand  bricks  will  cost  1.544  times  $7,  or  $10,808  -  $10.81. 

Rule. — Reduce  the  quantity  to  thousands  and  decimals  of  a  thousand, 
by  pointing  off  three  places  from  the  right,  then  multiply  by  the  cost  per  M. 

EXAaiPLES  rOK  PRACTICE. 

313.  1.     What  will  be  the  cost  of  1650  ft.  pine  lumber,  at  $15  per  M? 

2.  What  will  be  the  cost  of  611  ft.  oak  lumber,  at  $24  per  M? 

3.  What  will  be  the  cost  of  21 168  ft.  hemlock  lumber,  at  $7.50  per  M? 
Jf.     What  will  be  the  cost  of  9475  ft.  elm  lumber,  at  $13  per  M? 

5.  What  will  be  the  cost  of  2120  ft.  ash  lumber,  at  $25  per  M? 

6.  What  will  be  the  cost  of  2768  ft.  maple  lumber,  at  $14  per  M? 

7.  What  will  be  the  cost  of  1100  ft.  chestnut  lumber,  at  $18  per  M  ? 

8.  Find  the  cost  of  4560  ft.  oak  lumber,  at  $22  per  M. 

9.  Find  the  cost  of  11265  ft.  spruce  lumber,  at  $12.50  per  M. 

10.  Find  the  cost  of  6625  shingles,  at  $5.25  per  M. 

11.  A  dealer  bought  the  season's  cut  of  a  saw  mill,  which  was  as  follows: 
■326475  ft.  clear  pine,  at  $25  per  M;  1467250  ft.  seconds,  at  $17.50  per  M;  102500 
ft.  culls,  at  $13  per  M;  890000  ft.  hemlock  boards,  at  $10.50  per  M;  824650  ft. 
hemlock  timber,  at  $9  per  M;  552720  ft.  white  oak  plank,  at  $21  per  M;  75690 
ft.  red  oak  plank,  at  $16  per  M;  101145  ft.  cherry,  at  $35  per  M.  What  was  the 
amount  of  the  purchase  ? 

12.  For  constructing  a  house  and  barn  I  bouglit:  46210  ft.  matched  pine,  at 
$21  per  M;  13516  ft.  siding,  at  $28.50  per  M;  11260  ft.  chestnut,  at  $32  per  M; 
4680  ft.  black  walnut,  at  $45  per  M;  928  ft.  cherry,  at  ^^Q  per  M;  33725  ft. 
hemlock  timber,  at  $11  per  M;  58660  shingles,  at  $6.25  per  M;  13700  brick,  at 
-5.60  per  M.     What  was  the  total  cost  ? 

7 


98  EXAMPLES    FOR    PRACTICE. 

314.  To  find  the  Cost  of  Articles  Sold  by  the  Short  Ton,  or  Ton  of  2000  lb. 
Example. — What  will  be  the  cost  of  3108  lb.  of  coal,  at  $6  per  ton? 

ExPLAXATiOK.— 3108  lb.  =  3.108  thousand  lb.;  since  1  ton,  or  2000  lb.,  cost  $6,  i  ton,  or 
1000  lb.,  -will  cost  i  of  $6,  or  $3;  and  if  1000  lb.  cost  $3,  3.108  thousand  lb.  will  cost  3.108 
times  $3,  or  $9,324,  or  $9.32. 

Rule. — Diiide  the  price  of  one  ton  by  2,  and  the  result  wiU  be  the  price 
per  1000  lb.  From  the  right  of  the  quantity  point  off  3  places,  thus 
reducing  it  to  thousands  and  decimals  of  a  thousand.  Multiply  by  the 
price  per  1000  lb. 

EXAMPI.ES  FOK   PRACTICE. 

315.  1.     At  $3  per  ton,  what  will  be  the  cost  of  2680  lb.  soft  coal?' 
x\     At  87  per  ton,  what  will  be  the  cost  of  1345  lb.  canuel  coal? 

3.  At  $36  per  ton,  what  will  be  the  cost  of  4372  lb.  phosphate? 

4.  At  12.50  per  ton,  what  will  be  the  cost  of  11075  lb.  salt? 

5.  '  At  $34.50  per  ton,  what  will  be  the  cost  of  116780  lb.  pig  iron? 

6.  At  847.60  per  ton,  what  will  be  the  cost  of  84725  lb.  steel  rails? 

7.  At  ¥125  per  ton,  what  will  be  the  cost  of  15066  lb.  sheet  copper? 

8.  At  $4.50  per  ton,  what  will  be  the  cost  of  9362  lb.  land  plaster? 

9.  At  $2.10  per  ton,  what  will  be  the  cost  of  2640  lb.  slack  lime? 

10.  At  $35  per  ton,  what  will  be  the  cost  of  1115  lb.  giiauo? 

11.  What  will  be  the  freight,  at  $5  per  ton,  on  four  cars  of  Mdse.  of  21780, 
23055,  41200,  and  32460  lb.  weight  respectively? 

12.  At  $16.50  per  ton,  what  will  be  the  express  charges  on  five  boxes  weighing 
respectively  186,  610,  241,  519,  and  356  lb? 

13.  My  furnace  consumed,  in  one  year,  six  loads  of  hard  coal,  weighing  respec- 
tively 4125,  3960,  4305,  4440,  4055,  and  3775  lb.  If  the  coal  was  bought  at  $4.60 
per  ton,  what  did  it  cost  to  run  the  furnace  ? 

IJf.  A  dealer  stocked  his  yard  with  17500  tons  of  coal,  as  follows:  850  tons 
cannel,  at  $7.40  per  ton;  52600  lb.  soft,  at  82.50  per  ton;  193410  lb.  of  egg,  at 
$3.20  per  ton,  and  the  remainder  chestnut,  at  $3.60  per  ton.  What  was  the  value 
of  the  dealer's  stock  ? 

316.     To  Find  the  Cost  of  Products  of  Varying  Weights  per  BnsheL 

Example  1. — Required,  the  cost  of  104  lb.  of  clover  seed,  at  $6.35  per  bushel 
of  60  lb. 

ExPLAXATioK. — At  $6.35  per  lb.,  the  cost  would  be  104  times  $6.35,  or  $660.40;  but  since 
the  price  was  not  $6.35  per  lb.,  but  $6.35  per  bu.  of  60  lb.,  the  cost  will  be  ^V  of  $660.40,  or 
$11,006,  or  $11.01. 

Example  2. — Required,  the  cost  of  100  1b.  of  blue  grass  seed,  at  $1.25  per 
bushel  of  14  lb. 

ExPLAKATiox. — At  $1.25  per  lb.  the  cost  would  be  $125;  but  since  the  price  was  not  $1.25 
per  lb.,  but  $1.25  per  bu.  of  14  lb.,  the  cost  would  be  ^^  of  $125,  or  $8.93. 


EXAMPLES    FOR    PRACTICE.  99 

Jinie.— Multiply  the  number  of  pounds  weight  by  the  price  per  bushel, 
and  divide  the  product  hy  the  number  of  pounds  in  1  bushel. 

Remark.— Parts  of  bushels  are  often  written  in  smaller  figures  at  the  right  and  above  aa 
pounds.  Thus  1**  bu.  clover  seed  =  U^  bu.  =  1  bu.  44  lb.  =  104  lb.  21 1^  bu.  oats  =  21|| 
bu.  =  21  bu.  12  lb.  =  682  lb.     119««  bu.  corn  =  llOfl  bu.  =  119  bu.  25  lb.  =  7689  lb. 

EXAMPLES   rOK   PRACTICE. 

317.     How  much  should  be  paid  for  a  load  of 

1.  Wheat,  weighing  2142  lb.,  at  %  .80  per  bushel  of  GO  lb. 

2.  Corn,  weighing  2506  lb.,  at  S.G5  per  bushel  of  58  lb. 
S.     Barley,  weighing  3381  lb.,  at  $  .75  per  bushel  of  48  lb. 
Jf.     Millet,  weighing  1768  lb.,  at  $1  per  bushel  of  45  lb. 

5.  Oats,  weighing  2255  lb.,  at  1.35  per  bushel  of  32  lb. 

6.  Buckwheat,  weighing  2172  lb.,  at  8.60  per  bushel  of  48  lb. 

7.  Beans,  weighing  2761  lb.,  at  11.25  per  bushel  of  62  lb, 

8.  Peas,  weighing  2500  lb.,  at  $1.40  per  bushel  of  60  lb. 

9.  Hungarian  grass  seed,  weighing  3146  lb.,  at  $2,50  per  bushel  of  45  lb. 

10.  Eed  top  grass  seed,  weighing  2059  lb.,  at  $  .90  per  bushel  of  14  lb. 

11.  Timothy  seed,  weighing  2677  lb.,  at  %%  per  bushel  of  44  lb. 

12.  Kentucky  blue  grass  seed,  weighing  2266  lb.,  at  $1.50  per  bushel  of  14  lb. 

13.  Clover  seed,  weighing  2941  lb.,  at  $5.10  per  bushel  of  45  lb. 
IJf.  Flax  seed,  weighing  2727  lb.,  at  $2.25  per  bushel  of  56  lb. 

15.  Castor  beans,  weighing  3050  lb.,  at  $3  per  bushel  of  46  lb. 

16.  Potatoes,  weighing  2599  lb.,  at  $.65  per  bushel  of  60  lb. 
11.     Turnips,  weighing  2160  lb.,  at  $  .30  per  bushel  of  56  lb. 

18.  Apples,  Aveighing  2701  lb.,  at  $  .25  per  bushel  of  56  lb. 

19.  Sweet  potatoes,  weighing  3349  lb.,  at  $1  per  bushel  of  55  lb. 

20.  Onions,  weighing  2021  lb.,  at  $.85  per  bushel  of  57  lb. 

21.  Rye,  weighing  1367  lb.,  at  $  .64  per  bushel  of  56  lb. 

22.  The  products  of  a  farm  were  ten  loads  each  of  Avheat,  barley,  corn,  oats, 
and  potatoes.  The  wheat  sold  at  $1.12  per  bushel  of  60  lb.,  the  barley  at  85^ 
per  bushel  of  48  lb.,  corn  at  70^  per  bushel  of  58  lb.,  oats  at  32^  per  bushel  of 
32  lb.,  and  potatoes  at  629^'  per  bushel  of  60  lb.  The  loads  of  wlieat  weighed 
respectively  2585,  2640,  2721,  2594,  3063,  3354,  3145,  2720,  2938,  and  2890  lb.; 
the  barley  2163,  2487,  2225,  3004,  3121,  2742,  2907,  2525,  3140,  and  3082  lb.; 
the  corn  3100,  3126,  3097,  3040,  2872,  2950,  2777,  2981,  2547,  and  2939  lb.;  the 
oats  1973,  2946,  2172,  3148,  2500,  1951,  2631,  2997,  3005,  and  2775  lb,;  the 
potatoes  2846,  2891,  2805,  2863,  2984,  2901,  3046,  3280,  3395,  and  2584  lb. 
How  much  was  received  from  the  five  products? 

Remark. — Add  each  ten  loads,  and  compute  bushels  but  once  for  each  product. 


100 


BILLS,    STATEMENTS,    AXD    INVENTORIES. 


BILLS,  STATEMENTS,  AND    INVENTORIES. 


319.     A  Bill  is  a  \rritten 
rendered. 


statement  in  detail  of   articles   sold  or  setvices 


Remark. — A  Bill  should  state  the  names  of  both  parties,  the  terms  of  credit,  the  name, 
quantity,  and  price  of  each  item,  and  the  entire  amount.  The  Bill  is  said  to  l)e  receipted  when 
the  words  "  Received  Payment,"  or  "  Paid  "  and  the  creditor's  signature,  have  been  written  at 
the  bottom. 

3*20.  An  IiiToiee  is  a  written  description  of  merchandise  sold,  or  shipped  to 
be  sold  on  account  of  the  shipper. 

Remark  1. — The  terms  Invoice  and  Bill  are  now  used  interchangeably;  formerly  the  term 
Invoice  was  applied  only  to  written  statements  of  merchandise  shipped  to  be  sold  for  the  owner. 

2.  An  Invoice  should  bear  the  date  of  the  sale  or  shipment,  the  special  distinguishing 
marks,  if  any,  upon  the  goods,  the  names  of  seller  and  buyer,  or  consignor  and  consignee,  the 
items,  prices,  footing,  discounts,  if  any,  terms  of  sale,  and  manner  of  shipment. 

321.  A  Statement  is  based  upon  itemized  bills  previously  rendered,  and  is 
a  written  exhibit  of  the  sum  of  the  items  charged  in  each  of  the  bills,  including 
also  the  dates  on  which  the  several  bills  were  rendered. 

3*22.  An  luTeutory  is  an  itemized  schedule  of  the  property  possessed  by  an 
individual,  firm,  or  corporation,  and  not  shown  by  the  regular  books  of  account; 
or  it  may  include  all  of  the  property  possessed  by  an  individual,  firm,  or  corpo- 
ration, such  as  book  accounts,  notes,  cash,  merchandise.,  etc.,  and  also  the  debts 
due  by  the  individual,  firm,  or  corporation.  This,  however,  is  generally  called  a 
statement  of  the  business. 

Remark. — An  inventory  is  usually  made  upon  the  event  of  taking  off  a  balance  sheet,  of  a 
change  in  the  business,  of  the  admission  of  a  partner,  of  the  issue  of  stock,  or,  in  case  of 
embarrassment  or  insolvency,  for  examination  by  creditors,  together  with  the  other  resources 
and  liabilities  of  the  business. 


323.     Contractions  and  Abbreviations  used  in  Business. 


Al 

First  Quality. 

a.     Cent. 

E.  &  0.  E.     Error 

Acct 

Account. 

C7igd.     Charged. 

Omissions  Except 

Agt. 

Agent. 

Co.     Company. 

Exch.     Exchange. 

Amt. 

Amount. 

a    0.   D.     Collect   on 

Fol.     Folio  or  page. 

Bed. 

Balance. 

Delivery. 

Fr't.     Freight. 

Bbl 

or  Bar.     Barrel. 

Com.     Commission. 

Ft.     Foot. 

Bdl 

Bundle. 

Con.     Consignment. 

Gal.     Gallon. 

Blk. 

Black. 

Cr.     Creditor. 

Gr.     Gross. 

/l 

Bill  of  Lading. 

Cwt.     Hundred  weight. 

Guar.     Guaranteed 

Bot. 

Bought. 

Dft.     Draft. 

Hhd.     Hogshead. 

Bro. 

Brother. 

Dis.     Discount. 

i.  e.     That  is. 

Bu. 

Bushel. 

Do.  or  ditto.     The  same. 

In.     Inch. 

Bx. 

Box. 

Doz.     Dozen. 

Ins.     Insurance. 

Cd. 

Cord. 

Dr.     Debtor. 

Jr.     Junior. 

^    c 

ent. 

Ea.  ^  Each. 

Lb.     Pound. 

and 


BILLS. 


101 


Mdse.     Mercliandise. 

P.  or  p.     Page. 

Pec'd.     Received. 

Me7n.     Memorandum. 

Pp.  or  pp.     Pages. 

Rec't.     Receipt. 

Messrs.     Gentlemen  or 

Pat/'i.     Payment. 

R.  R.     Railroad. 

Sirs. 

Pd.     Paid. 

Schr.     Schooner. 

Mr.     Mister. 

Per.     By,  or  by  the. 

Ship't.     Shi])ment. 

Mrs.     Mistress. 

Pkf/.     Package. 

Str.     Steamer. 

N.  B.     Take  notice. 

P.  0.     Post  Office. 

Sunds.     Sundries. 

Net.     Without   discount. 

Pr.     Pair. 

Super.     Superfine. 

No.     Number. 

Pc.     Piece. 

Wt.     Weight. 

Oz.     Ounce. 

Qr.     Quarter. 

Yd.     Yard. 

Remark. — In  abbreviating  measures  of  capacity,  weight,  distance,  or  time,  it  is  unnecessary 
to  add  an  s  for  the  plural. 

324.     Time  Abbreviations  and  Contractions  used  in  Business. 


Jan.  or  Jajiy.     January. 

Nov.     November. 

Ce7it.     Century. 

Feb.  or  FeVy.     February. 

Dec.     December. 

d. 

Dav. 

Mar.     Marcli. 

Mo.     Month. 

It. 

Hour. 

Apr.     April. 

Yr.     Year. 

m. 

Minute. 

Aug.     August. 

Inst.     Present  month. 

sec. 

Second. 

Sept.     September. 

Prox.     Next  month. 

ick 

AVeek. 

Oct.     October. 

Ult.     Last  month. 

325.     Signs  and  Symbo 

Is  in  Common  Use. 

@     At;  as,  at  a  i)rico. 

"'^     Care  of. 

New  account. 

if     Number. 

y"     Check  mark. 

o/ 

Old  account. 

^     By,  or  by  the. 

;»     Per    cent,     or     Hun- 

X 

By,  in  surface 

^     Account. 

dredths. 

measures. 

BILLS. 
326.     Find  the  footing  of  each  of  the  following  bills: 


(1.) 


John  R.  Kxox, 

\rvi  Pearl  St.,  City, 


Knoxville,  Tenn,,  Dec.  31,  1888. 
Boufjht  of  CULVER  &   CASS. 


3 

sac 

2 

bu. 

i 

bu. 

2 

lb. 

2 

lb. 

1 

lb. 

2 

gal 

4 

bu. 

4 

lb. 

ks  Cream  Flour 95^- 

Potatoes 80i^' 

Sweet  Potatoes 90^' 

Ginger 22^'- 

Jap.  Tea 55^'- 

0.  H.  Tea. 7o^- 

.  Syrup 45^' 

Onions $1 

Crackers 11^- 

Paid, 


2  85 
1  60 
45 
44 
10 
75 
90 
50 
44 


Culver  &  Cass, 

Per  Cass. 


102 


Folio  246. 


BILLS. 


Saginaw,  Mich.,  Sept.  1,  1888. 
McGraw  &  Sage, 

Tonawanda,  N.  Y., 

To  WALLACE   W.  WESTON,  Dr. 
Terms,  Sight  Draft  without  notice  after  ninety  days;  5,'?  if  paid  within  60  days. 


26416  ft.  Clear  Pine 28.00  per  M. 

146250  ft.  Pine  Plank 23.50  per  M. 

81275  ft.  Clapboards 25.00  per  M. 

11670  Cedar  Posts 7.00  per  C. 

71300  Shingles  '^A" 4.10  per  M. 

56200  ft.  Pine  Timber 21.00  per  M. 

111224  Cedar  R.  R.  Ties. 34.50  per  C. 

91050  ft.  Flooring 27.50  per  M. 

25508  Shingles  "  B, " 3.60  per  M. 

31000  Barn  Boards 15. 75  per  M. 


(3.) 


Ole  Paulsen  &  Bro., 

Detroit,  Mich., 
Folio  41. 
Sales  Bk.  219. 

Terms  cash. 


Worcester,  Mass.,  May  15,  1888. 


To  FRANK   DRAKE   &   SON,  Dr. 


Case. 

1119 

15 

H    5 

12 

Pl 

0 

t    7 

24 

it  21 

21 

Pieces  Bleached  Cotton, 

412    403    411     452    44    441    471    453 

42     423     433    431     47    44    44^ 

Pieces  Muslin, 
371    323    33   353  341   32 
352    333    37   381   381   36 

Pieces  Delaine, 

39    402    411    393   3^2   40  423   44^    42 

Pieces  Windsor  Prmts, 

213  273  253  28    26  228  24  25  32  312 
28     241   25     272  22  281   24^22  21^26 
24     312  32     22 

Pieces  Merrimac  Prints, 

281   32     343  282  26    24i   222  242  262 
24    261  33     282  34   27i  30    323  24 
302  31     302 


No.i'd 


Price, 


7^- 
16^^ 

5i0 


Items. 


Amount. 


Remark. — Any  conditions  as  to  time  of  credit,  manner  of  payment,  interest  on  balance,  or 
discount  for  prepayment,  are  properly  placed  on  a  bill  or  statement. 


An  M  of  shingles  is  equiTalent  to  one  thousand  shingles  averaging  4  inches  in  width. 


STATEMENTS. 

(4.) 


103 


Book  3,  Page  308. 

H.  H.  Barnes  &  Co. 

Boston,  Mass., 

Terms,  Interest  after  sixty  dai/s. 


Chicago,  III,,  Aug.  1,  1888. 
Bought  of  PEASE   &   SONS. 


25 


baskets  Pork  Loins,  net 

312   301    297   315    302   313  8^^- 

tubs  Lard,  71-14   70-15    69-14 

pkg.  10^  each  11^ 

casks  Shoulders,  428-68   419-70 

423-65  432-72  pkg.  90^  each  9^ 

bar.  Mess  Pork  $22.50 


20    casks     Hams,         395-67    412-71  402-71 

411-67   408-68   425-71    400-69  399-70 

398-71   426-68   419-69   423-69  407-67 

415-75  418-68   409-71    403-71  421-71 

428-68  400-78  pkg.  75^  each  13i^- 


PkK. 


STATEMENTS. 

327.     Find  the  amount  of  each  of  the  following  statements  : 

(1.) 
Folio  1(21.  Birmingham,  Ala.,  Jan.  1.  1889. 

Richmond  &  New  Orleans  Railway  Co., 

To  CLIMAX   FOUNDRY   CO.,  Dr. 


1888. 

Nov. 

4 

To  Bill  r 

Bnde 

(. 

7' 

(( 

13 

(( 

18 

a 

21 

i( 

25 

a 

29 

<< 

30 

Dec. 

3 

t< 

6 

tt 

7 

(( 

10 

i( 

15 

a 

20 

i  ( 

21 

ii 

22 

ii 

25 

t( 

30 

Please  remit 

590 

25 

375 

13 

1150 

1560 

25 

2506 

50 

763 

28 

846 

20 

1000 

12750 

2634 

19 

9374 

75 

871 

03 

767 

20 

8500 

76 

50 

1438 

10 

119 

93 

1408 

27 

104 


STATEMENTS. 


(2.) 

Austin,  Texas,  Mar.  21,  1888. 
Geo.  H.  Grimes, 

Galveston,  Texas, 

In  acconnt  with  CLAUDE  M.  OGDEN,  Dr. 


1888. 

Jan. 

15 

a 

20 

(< 

24 

<( 

28 

Feb. 

1^ 

ti 

10 

K 

13 

(i 

18 

It 

20 

K 

22 

4( 

24 

(< 

29 

Feb. 

4 

ii 

27 

Mar. 

3 

(< 

15 

To  Bill  rendered 


N.  Y.  Dft. 
Cash, 


Or. 


Balance  due 


275 

41 

315 

07 

798 

10 

176 

42 

215 

84 

193 

76 

505 

75 

97 

22 

108 

47 

214 

29 

307 

62 

184 

36 

3392 

1200 

450 

275 

500 

2425 

967 

31 


31 


(3.) 


William  Warren, 

763  Madison  St.,  City, 


lUUO. 

Apr. 


Milwaukee,  Wis.,  June  12,  1890, 


Bought  of  HARRIS   BROS.  &  CO. 


2 

2 

O 

aJ 

2 

29 
29 
29 
29 
29 


2  pairs  Kip  Boots 3. 75 

2     "     Ladies' Shoes 4.25 

1     "     Child'sShoes 1.10 

1  doz.  Linen  Handk'ch'fs 1.80 

2  Neckties _ 35^- 

21  yd.  Dress  Silk 1.40 

46"     Bleached  Cotton 11^ 

15"     Muslin.. 12^^ 

5  "     Broadcloth 2.25 


Received  Payment, 

Harris  Bros.  &  Co., 

Per  L.  Harkis. 

Remark. — In  retail  business,  where  running  accounts  are  kept  with  customers,  a  transcript 
3f  the  charges,  or  of  charges  and  credits,  is  made,  giving  items,  dates  of  purchases  and  of  pay- 
aaeats,  and  so  partaking  of  the  nature  of  both  Statement  and  Bill. 


INVENTOKIES. 

INVENTORIES. 

328.     Find  the  amount  of  each  of  the  following  inventories 

(1.) 
Merchandise  Inventory,  J  ax.  l,  1888. 


105 


pc.  F.  A.  Cambric 

56  52  45  50  52  54  46  50^405 
gr.  Jet  Buttons, 
pc.  P.  D.  Goods 

55    453  552  503  51  52  461  50 

521   54    482  503  53  55150 
pc.   G.  Flannel 

353  40  402  403 
pc.  E.  Lining 

40  522  54  551  452  5o« 

pc.  V.  Barege 

201  05  232  27  263  22  242  22  2G3  28 
pc.  B.  H.  Checks 

45  52  55  41  402  513  511  53  508  46 
pc.  W.  Prints 

252  313  30  282  27 
pc.  A.  F.  Cashmere 

621  653  601  G3  583  6O2  562  558 

60    622  553  581603  58    55i 
pc.  L.  Gingham, 

45    481  461   442    453  443  46  44    48 

502  513  408   471   461  48    49  451  43 


22^ 
1.12i 

1 

50^ 

\ 

25^' 

34^- 

' 

mt 

1 

2i(/- 

H</: 

19^' 

46  42 

i 

(2.) 
Starbuck  &  Martin's  Inventory,  Jan.  1,  1889. 


Schedule  A.     {^Personal  Property. ) 

3  Delivery  Horses,  $110,  $95,  $165, 

4  "       Express  Wagons, 

3  "        Sleighs, 

4  sets  Single  Harness, 
Robes,  Blankets,  and  Whips, 
Grocery  stock,  as  by  Schedule  "G." 
Bills  receivable,  as  by      "         "H," 
Accts.       "  "        "         "  I," 
Fixtures  in  store,  movable. 

Schedule  B.     {Real  Estate. ) 

7  Vacant  Lots  on  Bank  St., 
3  Houses  on  Clayton  Pk., 

No.  12,  18,  and  20, 
Warehouse  on  Canal, 


/•) 

$80 

$35 

$12.50 

15 

1 

13246 

09 

7246 

25 

6242 

10 

975 

50 

$1250 

$2150 

13500 

106 


MISCELLANEOUS   EXAMPLES. 


MISCKLLANEOUS    EXAMPLKS. 

329.  1.  Maurice  H.  Decker,  bought  of  Silas  Kingsbury  &  Co.,  Elmira,  N.  Y., 
July  5,  1888,  17G0  ft.  pine,  at  $29  per  M  ;  40  cedar  posts,  at  $12.50  per  C;  nails 
and  hardware,  $G.21;  11248  ft.  stringers,  at  $4.75  per  M.  What  was  the  amount 
of  the  bill  ? 

2.  Geo.  W.  Banning,  bought  of  E.  B.  Henry  &  Co.,  Syracuse,  N.  Y.,  June 
13,  1888,  on  account,  2  doz.  carpet  stretchers,  at  $3;  10  grindstones,  at  $2.25;  5 
doz.  steelyards,  at  $9;  15  blacksmith  drills,  at  $T;  12  clothes  wringers,  at  $4.50; 
6  doz.  wrought  wrenches,  at  $12.25;  3  copying  presses,  at  $5;  7  doz.  cow  bells, 
at  $8.50;  15  doz.  cast  steel  axes,  at  $12.     Find  the  amount  of  the  bill. 

3.  Wm.  J.  Howard,  bought  for  cash  of  Howe  &  Collins,  carpet  dealers, 
Rochester,  N.  Y.,  July  1,  1888,  100  yd.  Moquette,  at  $1.75;  250  yd.  body  Brus- 
sels, at  $1.50;  325  yd.  tapestry  Brussels,  at  $1.00;  500  yd.  3-ply  ingrain,  at  75^; 
275  yd.  2-ply  ingrain,  at  65^;  300  yd.  matting,  at  25^;  200  yd.  lining,  at  12-J^'. 
How  much  money  was  recpiired  to  pay  the  bill  ? 

4.  Henry  R.  Smith,  bouglit  of  0.  L.  Warren,  Waverly,  N.  Y.,  Dec.  15,  1888, 
terms,  60da.;  2fi  off,  in  10  da. ;  3  doz.  Eagle  wash  boards,  at  $1.75;  5doz.  Novelty 
wash  boards,  at  $2.25;  5  M.  No.  4  paper  bags,  at  $1.75;  3  doz.  butter  bowls,  at 
$2;  5  doz.  0.  C.  trays,  at  $4;  1|  doz.  feather  dusters,  at  $18;  10  gro.  Gates' 
matches,  at  $2.75;  15  broom  racks,  at  $2.25;  5  doz.  wood  shovels,  at  $7.50;  |-  doz. 
oil  tanks,  at  $16.     What  was  the  amount  of  the  bill  ? 

J.     Jeffrey  &  Co.,  bought  of  Perry  &  Co.,  Buffalo,  N.  Y.,  Sept.  1,  1888: 


10  pc.  F.  of  L.  cotton,  50  60^  65^  51  60 

55  52  62  61  56,  at  8^. 
5  doz.  C.  silk,  at  80^ 

4  pc.  A.  F.  cashmere,  62^  51  ^  55  60,  at 
19^. 

5  pc.  A.   L.   L.  cotton,  40  46^  51  ^   55 
42S  at  4^. 

500  lb.  W.  S.  warp,  at  15^'. 
Find  the  amount  of  the  bill. 


10  pc.  M.  shirting,  40  41  46  34  51  45  50 

43  52  42,  at  o(f-. 
15  pc.  crash,  600  yd.,  at  5^. 
6  pc.  C.  jeans,  SO^  45^  50  55  61^  46,  at 

10  doz.  M.  L.  thread,  at  59^. 

10  pc.  R.  print,  41  55  45  51  46  50  40  66 

42  52,  at  U(/: 


6.     W.  C.  Blanchard,  bought  of  M.  C.  Wood,  Utica,  N.  Y.,  July  15,  1888: 


10  i)C.   R.  gingham,  60  61  ^  50^  60^  51 

613  61  50  55  513,  at  8(f: 
10  doz.  F.  E.  braid,  at  23^-. 
10  pc  B.  checks,  45  41  55 1  42  52  40^ 

50  55  513  452^  at  24^-. 
15  gro.  G.  buttons,  at  $1.12^. 
2  pc.  T.  A.  flannel,  65  60,  at  30^-. 
6  pc.  E.  lining,  40  55 1  452  52  41  501, 

at  5<f: 
5  doz.  L.  L.  gloves,  at  $3.05. 

What  was  the  amount  of  the  bill  ? 


4  pc.  N.  sateen,  553  55  50  eo^,  at  5^^. 

5  gross  T.  Braid,  at  $7. 62|. 

3  doz.  L.  shirts,  at  $7.20. 

6  pc.  T.  R.  print,  25  35  303  31  21  25 1, 
at  4|^. 

10  cases  E.  Batts,  at  $6.00. 
20  gro.  S.  P.  buttons,  at  49^-. 

4  pc.  V.  barege,  20,  23  25  25,  at  16^. 

7  pc.  W.  Print,  453  51  45  50  462  55  50^ 
at  5i^. 


MISCELLANEOUS   EXAMPLES. 


107 


7.     I.  F.  Hoyt,  bought  of  Mann  &  Moore,  Sept.  4,  1888,  terms  30  da. : 


10  pc.  X.  sateen,  oo^  51  50^  54i  5G   55 

522  53  513  50,  at  5|^'. 
15  pe.  T.  A.  flannel,  62^  65 ^  61  58^  55 

631  653  62  602  6,3  sgs  ^^i  53  623  65^ 

at  33^^'. 

What  was  the  footinjr  of  the  bill  ? 


20  pc.  R.  Gingham,  50  52 1  51  51 2  55 
603  621  612  58  552  5gi  533  51  553 
612  61  581  56  542  511^  at  6i^. 

10  pc.  B.  checks,  45  52 1  412  40  55^  50* 
45  511  43  503,  at  25^/-. 


S.     H.  B.  Smith,  bought  of  Jones  Bros.  &  Co.,  Dec.  3,  1888: 


19  pc.  M.  gingham,  472  36  41^  491  39^ 
41  323  34  361  433  46  353  331  45  50 
483  332  391  36,  at  11/. 

20  pc.  P.  B.  sheeting,  323  331  372  40 
Find  the  footing  of  the  bill. 


441  443  51  402  392  373  35  382  35  41* 
463  492  381  413  382  361,  at  6|/ 
10  pc.  B.  D.  velvet,  212  273  25  262  293 
222  243  21  203  232,  at  $6.50. 


9.     Drown  Bros.  &  Co.,  bought  of  W.  B.  Adams  &  Co.,  for  cash,  June  18,  1888: 


20  pc.  L.  gingham,  582  451  413  331  462 

453  512  55  382  35  373  493  402  513  44 

442  40  371  333  462,  at  ^<f;. 
24  pc.  ^\.  print,  44i  463  513  393  412  45 

483  51  343  372   35  362  413  343  491 

What  sum  of  money  Avas  required  to  pay  the  bill  ? 

10.     Find  the  amount  of  the  following  inventory: 


372  34  362  423  48  432  531  331  42,  ^t 

Hi-- 

20  pc.  E.  lining,  45  54i  392  483  462  332 
471  372  453  463  424  443  453  431  352 
542  343  422  533  441^  at  4i/. 


25  pc.  M.  gingham,  462  432  391  473  41 
50  393  503  42  443  362  34^  361  492 
403  413  392  401  493  45  383  33  382 
462  321,  atlOi/. 

40  pc.  L.  gingham,  35  362  333  411  33 
401  353  382  46  482  43  343  45  39  331 


39  432  472  42  362,  at  8/. 
15  pc.  E.  lining,  47  413  49  502  46  451 

383  36  412  381  453  33  402  391  45^  ^t 

UJ. 
10  pc.  L.  plaid,  462  431  331  353  401 

383  41  322  363  35^  ^t  !()<}■. 


373  342  48  362  32  383  471  50  482  41i  4pe.  C.denims,  392  6I1  483  362  ^t  12^/. 
351  39  423  44  412  451  48  433  36  33i 


108  DE>fOMI>"ATE   NUMBERS. 


DENOMINATE    NUMBERS. 

330.  Denominate  numbers  may  1>e  either  simple  or  compound. 

331.  A  Simple  Denominate  Nnmber  is  a  unit  or  a  eollection  of  units  of 
but  one  denomination. 

332.  A  Componnd  Denominate  Nnmber  is  a  concrete  mimher  expressed  in 
two  or  more  different  denominations;  as  5  lb.  4  oz,  12  dr.;  4  yr.  7  mo.  12  da. 

Remark. — Compound  denominate  numbers  are  sometimes  called  compound  numbers. 

333.  Componnd  Numbers  express  divisions  of  time,  and  of  the  money, 
weights,  and  measures  of  the  different  countries. 

Remark. — Most  denominate  scales  are  varying,  but  the  uniform  decimal  scale  i?  used 
throughout  the  metric  system,  and,  except  in  Great  Britain,  in  the  money  of  most  civilized 
countries.     The  units  oi  all  denominate  numbers  are  treated  by  the  decimal  scale. 

334.  A  Denominate  Fraction  is  a  fraction  expressing  one  or  more  of  the 
equal  parts  of  a  denominate  or  concrete  unit;  as  f  of  a  ton,  4  of  a  yd.,  ^  of  a  gal. 

335.  Reduction  of  Denominate  Numbers  is  the  process  of  changing  them 
from  one  denomination  to  another,  without  altering  their  value.  It  is  of  two 
kinds.  Reduction  Descending  and  Rednction  Ascending. 

336.  Reduction  Descending  is  the  process  of  changing  a  denominate  num- 
ber to  an  equivalent  number  of  a  lower  denomination;  as  the  change  of  barrels  to 
an  equivalent  in  gallons,  quarts,  pints,  or  gills. 

337.  Reduction  Ascending  is  the  process  of  changing  a  denominate  num- 
ber to  an  efjuivalent  of  a  higher  denomination;  as  the  change  of  gills  to  an 
equivalent  in  pints,  quarts,  gallons,  or  barrels. 


MEASURES   OF    TIME. 

338.  Time  is  the  measure  of  duration  ;  its  computations,  being  based  upon 
planetary  movements,  are  the  same  in  all  lands  and  among  all  peoples. 

339.  The  Solar  Day  is  the  unit  of  time;  it  includes  one  revolution  of  the 
earth  on  its  axis,  and  is  divided  into  -iA:  hours,  counting  from  midnight  to 
midnight  again. 

340.  Noon,  marked  M.  for  Meridian,  is  that  moment  of  time  at  which  a 
line,  called  a  Meridian,  projected  from  the  centre  of  the  earth  to  the  sun,  would 
pass  through  the  point  of  observation. 

341.  A.  M.  {Ante- Meridian)  denotes  the  12  hours  before  noon. 


MEASURES   OF   TIME.  109 

342.  p.  M.  {Post- Meridian)  denotes  the  time  between  noon  and  the  follow- 
ing midnight. 

Remarks.— 1.  For  astronomical  calculations,  the  day  begins  at  12  o'clock  noon,  but  for 
civil  affairs,  it  begins  at  12  o'clock  midnight. 

2.  In  banking  business,  the  law  fixes  the  end  of  the  day  at  the  hour  appointed  for  closing 
the  bank. 

34-3.  The  Solar  Year  is  the  exact  time  required  by  the  earth  to  make  one 
complete  revolution  around  the  sun.  It  is  equal  to  365  days,  5  hours,  48  minutes, 
49.7  seconds,  nearly  365^  days. 

344.  The  Common  Year  consists  of  365  days  for  3  successive  years;  and 
exery  fourth  year,  except  it  be  a  centennial  year,  contains  366  days,  one  day  being 
added  for  the  excess  of  the  solar  year  over  365  days;  this  day  is  added  to  tiie 
month  of  February,  which  then  has  29  days,  and  the  year  is  called  LeajJ  Year. 
The  slight  error  still  existing  after  this  addition,  is  again  corrected  by  excluding 
from  the  leap  years  the  centennial  years  which  are  not  divisible  by  400.  Thus 
1900,  2100,  2200,  while  divisible  by  4,  are  not  divisible  by  400,  hence  will  not  be 
leap  years;  while  2000,  2400,  2800,  being  divisible  by  400,  will  be  leap  years. 

Remarks. — 1.  The  correction  last  named  was  made  by  a  decree  of  Pope  Gregory  XIII.,  in 
1685,  and  is  known  as  the  Gregorian  calendar.  It  is  used  in  all  civilized  countries  except 
Russia,  and  is  so  nearly  correct  that  an  error  of  one  daj'  will  not  be  shown  for  4000  years, 
hence  it  is  practically  correct. 

2.  The  calendar  in  general  use  previous  to  1685  was  known  as  the  Julian  calendar,  having 
been  established  by  Julius  Caesar,  46  B.  C.  This  calendar  is  still  in  use  in  Russia,  and  as  the 
difference  in  the  two  calendars  is  now  12  days,  the  current  date  in  Russia  is  12  days  behind 
that  of  the  other  civilized  countries  of  the  world;  thus  when  it  is  Jan.  1  in  Russia,  it  is  Jan. 
13  in  all  other  countries. 

3.  The  Julian  and  the  Gregorian  calendars  are  sometimes  designated  by  the  terms  Old  Style 
(0.  S.),  and  New  Style  (N.  S.) 

345.  Rule  for  Leap  Years.— I-  All  years  divisible  by  4>  e-vcept  cen- 
tennial years,  are  leap  years. 

n.    JJl  centennial  years  divisiUe  hy  400  are  leap  years. 

Table. 

60  seconds  (sec. )       =  1  minute min. 

60  minutes  =  1  hour hr. 

24  hours  —  1  day da. 

7  (lays  =  1  loeek wh. 

Jf  weeks  =  1  lunar  month mo. 

30  days  :=  1  commercial  month  . .     mo. 

565  days  =  1  common  year yr. 

566  days  =  1  leap  year yr. 

12  calendar  months  =  1  civil  year yr. 

10  ijears  =  1  decade 

100  years  =  1' century C. 

Scale,  descending,  12,  30,  24,  60,  60;  ascending,  60,  60,  24,  30,  12. 

Remark.— In  most  business  transactions  30  days  are  considered  a  month,  and  twelve  such 
•months  a  year. 


110 


REDUCTION    OF   TIME. 


7th. 

July  (July)           having  31  days. 

8th. 

August  (Aug.)         " 

31    - 

9th. 

September  (Sept.)  " 

30    '' 

10th. 

October  (Oct.) 

31    " 

nth. 

November  (Nov.)    " 

30    " 

12th. 

December  (Dec.)     " 

31    '' 

346.  The  Calendar  Months  are  as  folk 
1st.  January  (Jan.)  having  31      days. 

2nd.  February  (Feb.)     "       28-29  " 

3rd.  March  (Mar.)         "       31 

4th.  April  (Apr.)  ''       30 

5th.  May  (May)  "       31 

6th.  June  (June)  "       30 

347.  The  year  begins  with  the  first  day,  or  First,  of  January,  and  is  divided 
into  four  seasons  of  three  months  each. 

348.  The  Seasons  are  Winter,  Spring,  Summer,  and  Autumn,  or  Fall. 
The  Winter  montiis  are  December,  January,  and  February. 

The  Si)ring  months  are  March,  April,  and  May. 

The  Summer  months  are  J^ine,  July,  and  August. 

The  Autumn  months  are  September,  October,  and  November. 

Remark.— The  ancient  Roman  year  began  with  March  1,  and  thus  September,  October, 
November,  and  December  ranked,  as  their  Latin  derivation  indicates,  as  the  7th,  8th,  9th,  and 
10th  months  respectively  of  the  Roman  year. 


REDUCTION   OF  TIME. 

349.  Tiie  reduction  of  expressions  of  time  from  higher  to  lower  denomina- 
tions, or  the  reverse,  may  be  accomplished  in  the  same  manner  as  the  reduction 
of  United  States  money  heretofore  explained,  the  only  difference  being  that  the 
scale  in  the  latter  is  uniform,  Avhile  that  in  the  former  is  varying. 


350.    To 

Example, 


3  vr, 


Operation. 
mo.  11  d.  7  hr. 


12 


30  mo. 
7  mo. 

43  mo. 
30 


1290  da. 
11  da. 


1301  da. 
24 


31224  hr. 
7hr. 

31231  hr. 


Reduce  Time  from  Higher  to  Lower  Denominations. 

— Reduce  3  yr.  7  mo.  11  da.  7  hr.  25  m.  38  sec.  to  seconds. 

Explanation. — Since    one    year 
25m.  38  sec.  equals  12  months,  3  years  equal  36 

months,  and  7  months  added  gives 
43  months;  since  one  month  equals 
30  days,  43  months  equal  1290  days, 
and  11  days  added  gives  1301  days; 
since  one  day  equals  24  hours,  1301 
days  equal  31224  hours,  and  7  hours 
added  gives  31231  hours;  since  one 
hour  equals  60  minutes,  31231  hours 
equal  1873860  minutes,  and  25  min- 
utes added  gives  1873885  minutes; 
since  one  minute  equals  60  seconds, 
1873885  minutes  equal  112433100 
seconds,  and  38  seconds  added  gives 
112433138  seconds. 


Operation  Continued. 
31231  hr. 
60 

1873860  m. 
25  m. 


1873885  m. 
60 

112433100  sec. 
38  sec. 

112433138  sec. 


Remark. — The  reduction  descending  of  any  compound  denominate  number  can  be  accom- 
plished as  above,  by  observing  the  scale  of  the  table  to  which  it'belongs. 


ADDITION   OF   TIME.  HI 

jj^nlg, Beginning  with  the  highest,  multiply  the  units  of  each  denoiyv- 

ination  hy  the  numher  in  the  scale  required  to  reduce  it  to  the  denoin- 
ination  next  lower;  add  the  units,  if  any,  of  such  lower  denomination, 
and  so  continue  from  the  given  to  the  required  denortvination. 

351.     To  Reduce  Time  from  Lower  to  Higher  Denominations. 

Example.— Reduce  1124:33138  seconds  to  years. 
Opekation.  " 

60  )  112433138  sec.  Explanation.— Divide  the  given 

.  seconds  by  60,  to  reduce  to  minutes; 

60  )  1873885  mm.  +  38  sec.  ^^^  minutes  thus  obtained,  by  60,  to 

24  )  31231  hr.  +  25  min,  reduce  to  hours;  the  hours  by  24,  to 

3oTl301  d'l  +  7  hr  reduce  to  days;  the  days  by  30,  to 

*  '  reduce  to  months,  and  the  months 

12J_43  mo.  +  11  da.  ^ij  12,  to  reduce  to  years, 

3  yr.      4-  7  mo. 
112433138  sec.  =  3  yr.  7  mo.  11  da.  7  hr.  25  min.  38  sec. 

Rule. — Divide  the  given  units  hy  that  numher  in  the  scale  ivhich  will 
reduce  them  to  units  of  the  next  higher  denomination,  and  so  continue 
from  the  given  to  the  required  denomination.  Any  remainder  ohtained 
will  he  of  the  same  denomination  as  the  dividend  from  which  it  arises. 


ADDITION   OF   TIME. 
352.     To  Add  Time. 

Time  expressions  may  be  added  as  simple  numbers,  if  only  it  be  observed  that 
the  scale  from  the  lowest  to  the  higest  order  is  60,  60,  24,  30,  and  12.  The 
highest  denomination  in  common  use  is  the  year. 

Example.— Add  41  yr.  8  mo.  22  da.  19  hr.  27  min.  14  sec,  and  5  yr.  .6  mo. 
11  da.  10  hr.  50  min.  56  sec. 

Exp  LAN  A  T  I  o  N .  — Arrange  the 
27  min.  14  sec.  numbers  so  that  those  of  the  same 
50  min       56  sec      denomination  stand  in  the  same  ver- 

! '.     tical    line.     Then    begin   with    the 

47  yr.  3  mo.  4da.  6  lir.  18  min.  10  sec.  lowest  denomination,  which  is  sec- 
onds, and  add:  14  seconds  plus  56  seconds  equals  70  seconds,  equals  1  minute  plus  10  seconds; 
write  the  10  underneath  the  column  of  seconds,  and  carry  the  1  to  the  next  column;  27  minutes 
plus  50  minutes  equals  77  minutes,  and  77  minutes  plus  1  minute  (to  carry)  equals  78  minutes, 
equals  1  hour  plus  18  minutes;  write  and  carry  as  before;  19  hours  plus  10  hours  equals  29  hours, 
and  29  hours  plus  1  hour  (to  carry)  equals  30 hours,  equals  1  day  plus  6  hours;  22  days  plus  11  days 
equals  33  days,  and  33  days  plus  1  day  (to  carry)  equals  34  days,  equals  1  month  plus  4  days; 
8  months  plus  6  months  equals  14  months,  and  14  months  plus  1  month  (to  carry)  equals  15 
months,  equals  1  year  plus  3  months;  41  years  plus  5  years  equals  46  years,  and  46  years  plus 
1  year  (to  carry)  equals  47  years. 

Hula.— Add  as  in  ahstract  numl)ers,  and  reduce  according  to  the  table 
of  Time. 


Operation. 

41  yr. 

8  mo. 

22  da.     10  hr. 

5  yr. 

6  mo. 

11  da.     10  hr. 

11*2  SUBTKACTION   OF  TIME. 

SUBTRACTION    OF   TIME. 

353.  Difference  iu  time  is  found  in  two  ways: 

1st.  B)-  counting  the  actual  number  of  days  from  the  given  to  the  required 
•date.  Thus,  the  number  of  days  between  May  13  and  September  7  is  117,  count- 
ing IS  days  left  in  May,  30  for  June,  31  for  July,  31  for  August,  and  the  7  of 
September. 

2d.  By  Compound  Subtraction.  Subtraction  in  either  simple  or  compound 
numbers  is  really  the  same,  except  that  in  the  latter  a  varying  scale  is  employed. 
That  is,  it  may,  and  usually  does,  involve  a  transformation  in  either  case.  This 
will  always  be  required  unless  the  several  minuend  terms,  or  orders  are  each  equal 
to  or  greater  than  the  corresponding  subtrahend  term. 

354.  To  Find  the  Difference  in  Time  by  Compound  Subtraction. 

Example. — Subtract  5  yr.  4  mu.  'il  da.  from  S  yr.  1  mo.  18  da. 

Operatiok.  Explaxatiox. — Write  the  numbers  so  that  those  of  the 

S  vr.  1  mo.  IS  da.  same  denomination  stand  iu  the  same  column.  Then  begin 
-5  vr         4  mo         21  da         ^^^^  ^^^  lowest  denomination  to  subtract.     Since  21  days  can- 

-^ not  be  subtracted  from  18  days,  transform,  or  borrow  one  from 

2  yr.  8  mo.  27  da.  the  next  denomination;  1  month  ;;::  30  days,  and  18 days  added 
=  48  days ;  48  days  —  21  days  =  27  days,  which  write  underneath  the  column  of  days ;  the  1  month 
having  been  borrowed  from  the  minuend,  there  are  no  months  remaining  from  which  to  sub- 
tract the  4  months  in  the  sul)trahend,  hence,  borrow  one  from  the  next  denomination;  12 
months  —  4  months  =  8  mouths,  which  write  underneath  the  column  of  months;  there  now 
remains  7  years  from  which  to  subtract;  7  years  —  5  years  =  2  years,  which  write  imderneath 
the  column  of  years.  This  completes  the  operation,  giving  a  remainder  of  2  years,  8  months, 
.and  27  days. 

Rule. — Snhtract  as  in  abstract  mnubcrs,  ohseri-ing  the  i-arying  scale. 

EXA3IPI.ES  FOK   PRACTICE. 

Remark. — In  the  following  examples,  the  difference  in  time  should  be  found  by  compound 
subtraction,  unless  it  be  otherwise  stated. 

355.  1.     Reduce  2T051  seconds  to  minutes. 

2.  Reduce  83129  seconds  to  hours  and  minutes. 

3.  Reduce  610251  seconds  to  higher  denominations. 

4.  How  many  years,  months,  days,  hours,  and  minutes,  in  749520360  seconds? 
0.  How  many  hours  from  half-past  three  o'clock  p.  m.  Oct.  13,  1888,  to  noon 

on  the  fourth  day  of  July,  1S89? 

6.  A  note  entitled  to  93  days'  time  was  dated  Oct.  13,  1888.     Counting 
actual  time,  on  what  day  should  it  be  paid? 

7.  How  many  days  between  Nov.  3,  1890,  and  Mar.  1,  1900? 

8.  A  mortgage  dated  July  2,  1888,  was  paid  Sept.  14,  1891.     How  many 
days  did  it  run? 

9.  How  long  does  a  note  run  if  dated  Sept.  22,  1887,  and  paid  Aug.  31,  1888? 
10.     How  much  time  will  a  man  gain  for  labor  in  60  years,  by  rising  45 

minutes  earlier  each  day,  beginning  Jan.  1,  1888. 


LATITUDE,    LONGITUDE,    AND   TIME.  113 

11.  How  many  more  minutes  in  the  eleven  years  before  Jan.  1,  1890,  than  in 
the  eleven  years  after  that  date  ? 

12.  How  many  seconds  of  difference  in  the  time  of  one  solar  year  and  12 
lunar  months  of  29  da.  12  hr.  44  min.  and  3  sec.  each  ? 


CIRCULAR   MEASURE. 

356.  Circular  Measure  is  used  in  surveying,  navigation,  astronomy,  and 
geography;  for  reckoning  latitude  and  longitude,  determining  location  of  places 
and  vessels,  and  in  computing  differences  of  time. 

357.  Every  circle,  great  or  small,  is  divisible  into  four  equal  parts;  these  parts 
are  called  quadrants,  and  are  divisible  into  ninety  equal  parts,  each  of  which  is 
called  a  degree;  every  circle,  therefore,  may  be  divided  into  360  equal  parts,  called 
degrees. 

Remark. — The  divisions  into  twelfths  called  signs,  and  into  sixths  called  sextants,  are  in 
occasional  use. 

Table. 

60  seconds  (")  =  1  minute  (').  30  degrees  —  1  sign  {S.) 

60  minutes       =  1  degree    {).  12  signs  or  360°  =  1  circle  (C) 

q    ,       j  descending,  12,  30,  60,  60;  or,  360,  60,  60. 
^^^^®'    \  ascending,  60,  60,  30,  12;  or,  60,  60,  360. 

Remark. — Minutes  of  the  earth's  circumference  are  called  nautical  or  geographic  miles. 

EXAMPLES   FOK   PRACTICE. 

358.  1.     Reduce  2154'  to  degrees. 

2.  Reduce  87406"  to  degrees,  minutes,  and  seconds. 

3.  Reduce  330581"  to  higher  denominations. 

4.  How  many  seconds  in  a  circle? 

o.     How  many  minutes  in  2  S.  21°  47'? 
6.     How  many  seconds  in  1  S.  27°  8'  57"? 
Reduce  8162  geographic  miles  to  degrees. 

How  many  geographic  miles  in  the  circumference  of  the  earth? 
By  two  different  observations  the  position  of  a  ship  was  shown  to  have 
•changed  519  geographic   miles.     How  much  was  her  change  in  degrees  and 
minutes? 


LATITUDE,  LONGITUDE,  AND  TIME. 

359.  Latitude  is  distance  north  or  south  from  the  equator.  A  place  is  said 
to  be  in  north  latitude  if  north  of  the  equator;  and  to  b^  in  south  latitude  if  south 
of  the  equator. 

360.  Longitude  is  distance  east  or  west  from  any  given  starting  point  or 
meridian.  A  place  is  said  to  be  in  west  longitude  if  west  of  the  given  meridian; 
and  to  be  in  east  longitude  if  east  of  the  given  meridian. 

8 


114  LATITUDE,    LONGITUDE,    AND   TIME, 

361.  Since  every  circle  may  be  divided  into  360  equal  parts,  or  degrees,  and 
the  sun  appears  to  pass  from  east  to  west  around  the  earth,  or  through  360°  of 
longitude,  once  in  every  24  hours,  it  will  pass  through  -^  of  360°,  or  15°  of  longi- 
tude, in  1  hour;  through  1°  of  longitude  in  ^  oi  I  hour,  or  4  minutes;  and 
through  1'  of  longtitude  in  -gV  of  -J^  minutes,  or  4  seconds. 

Table. 

360°  of  longitude  =  '^4  hours  or  1  day  of  time, da, 

15°"         "  =     1  hour  of  time,-- hr. 

1°  ♦•         *'  =    4  minutes  "         -. min. 

1'  '•         •'  =    4  seconds    '"         ..-    sec. 

Remark. — Standard  Time. — Previous  to  1883  there  were  fifty-three  different  time- 
staodards  in  use  by  the  railroads  of  the  United  States,  and  as  these  standards  were  based  on 
the  local  time  of  the  principal  cities  which  served  as  the  center  of  operations  of  the  different 
roads,  they  were  a  constant  source  of  annoyance  and  trouble,  lx)th  to  the  railroads  and  to  the 
traveling  public.  To  obviate  this  difficulty  the  principal  railroads  of  the  United  States  and 
Canada  adopted,  in  1883,  what  is  known  as  the  "Standard  Time  System."  This  system  di\ides 
the  United  States  and  Canada  into  four  sections  or  time- belts,  each  covering  15'  of  longitude, 
7^°  of  which  are  east  and  7^'  west  of  the  governing  or  standard  meridian,  and  the  time 
throughout  each  belt  is  the  same  as  the  astronomical  or  local  time  of  the  governing  meridian 
of  that  belt.  The  governing  meridians  are  the  Toth,  the  90th,  the  105th  and  the  120th  west  of 
Greenwich,  and  as  these  meridians  are  just  15  apart,  there  is  a  difference  in  time  of  exactly 
one  hour  between  any  one  of  them  and  the  one  next  on  the  east,  or  the  one  next  on  the  west; 
the  standard  meridian  next  on  the  east  being  one  hour  faster,  and  the  one  next  on  the  west  one 
hour  slower.  The  time  of  the  T5th  meridian,  which  is  about  4  minutes  slower  than  New  York 
time  and  about  1  minute  faster  than  Philadelphia  time,  is  called  '"  Eastern  Time,"  and  when  it 
is  astronomical  noon  on  this  meridian  it  is  noon  on  every  railroad  clock  from  Portland,  Me.,  to 
Buffalo  and  Pittsburg,  and  from  Quebec  to  Charleston.  The  time  of  the  90th  meridian,  one 
hour  slower  than  "  Eastern  Time,"  and  9  minutes  slower  than  Chicago  time,  is  known  as 
"  Central  Time,"  and  aU  roads  operated  in  the  second  belt  are  run  by  "  Central  Time."  The 
time  of  the  10.5th  meridian,  one  hour  slower  than  "  Central  Time,"  is  distinguished  as  "  Moun- 
tain Time."  Time  in  the  fourth  belt,  which  is  governed  by  the  120lh  meridian,  and  extends 
to  the  Pacific  coast,  is  c-alled  "Pacific  Time;"  it  is  one  hoiu*  behind  "Mountain  Time,"  two 
behind  "Central  Time,"  and  three  behind  "Eastern  Time.'"  The  changes  from  one  time- 
standard  to  another  are  made  at  the  termini  of  road,*,  or  at  well-known  points  of  departure, 
and  where  they  are  attended  with  the  least  inconvenience  and  danger.  As  this  system  has 
produced  satisfactory  results  and  has  been  adopted  by  most  of  the  principal  cities  for  local 
use,  it  is  probable  that  the  business  of  the  whole  country  will,  before  many  years,  be  regulated 
by  standard  railroad  time. 

362.  To  Find  the  Difference  in  Time,  when  the  Difference  in  Longitude  is 
given. 

Example. — If  the  difference  in  longitude  of  two  places  be  9*^  15',  what  must 
be  their  difference  in  time  ? 

Operation.  Explanation. — Since  each  minute  of  distance  equals  4  seconds  of 

Qo    I    1  =/  time,  15  minutes  of  distance  will  equal  15  times  4  seconds,  or  60  seconds, 

which  equals  one  minute  of  time.     And  since  each  degree  of  distance 

equals  4  minutes  of  time,  9  degrees  will  equal  9  times  4  minutes,  or  36 

37   min.      0  sec.      minutes;  adding  the  one  minute  obtained  above,  gives  37  minutes  as  the 
required  result. 


LATITUDE,    LONGITUDE,    AXD   TIME.  115 

Rule.— Multiply  the  units  of  distance  hy  Jj.,  and  reduce  according  to 
the  table  of  Time. 

EXAMPLES   FOR   PKACTICE. 

Remark. — Examples  under  this  topic  will  be  restricted  to  variations  of  solar  time. 

363.  1.  Cincinnati  is  84°  24',  and  San  Francisco  122°,  west  lonsritudc  What 
is  their  difference  in  time? 

2.  New  York  is  74°  1',  and  Halifax  63°  30',  west  longitude.  Find  tlieir 
difference  in  time. 

3.  St.  Petersburg  is  30°  19'  east,  and  St.  Louis  90°  15'  west  longitude  When 
it  is  noon  at  St.  Petersburg,  what  is  the  time  at  St.  Louis. 

Hemakk. — If  one  place  be  east  and  the  other  west  of  the  given  meridian,  to  find  their 
difference  in  longitude,  add  their  respective  distances  from  the  meridian  taken. 

Jf.  The  longitude  of  the  City  of  Mexico  is  99°  5',  and  that  of  Boston  71°  3', 
west  longitude.     Find  their  difference  in  time. 

5.  If  on  leaving  London,  0°  0'  of  longitude,  my  watcli,  keeping  correct  time, 
indicates  46  minutes,  15  seconds,  after  3  P.  m.,  what  time  should  it  indicate  on 
my  arrival  at  Astoria,  Oregon,  124°  west,  where  it  is  then  noon? 

364.  To  Find  the  DiflFerence  of  Longitude,  when  the  Difference  in  Time  is 
Given. 

Example. — The  difference  in  time  between  two  places  is  2  hours,  19  minutes, 
and  48  seconds.     What  is  their  difference  of  longitude? 

Operation.  Explanation. — 2  hours,  19  minutes,  and 

2  hr.  19  min.  48  sec.  —  139  min.  48  sec.      *?  seconds  equal  139  minutes  and  48  seconds; 
.  V  ..  _  _       .  since  each  4  minutes  of  time  equal  1  degree 

4  )  139  mm.  48  sec.  ^^  distance,  139  minutes  and  48  seconds  equal 

34°  +  (3  min.  48  sec. )  34  degrees,  with  3  minutes  and  48  seconds,  or 

3  min    48  sec   =  2*^8  sec  228 seconds,  remainder;  and  since  each  4  sec- 

onds of  time  equal  1'  of  distance,  228  seconds 

4  )  228  sec.  equal  57'  of  distance.     Therefore,  if  the  dif- 

57'  ference  in   time   between  two  points   be    2 

2  hr.  19  min.  48  sec   =  34°  57'  hours,  19  minutes,  and  48  seconds,  their  dif- 

ference in  longitude  will  be  34°  57'. 

Kule. — Reduce  the  difference  in  time  to  luinutes  and  seconds,  and 
divide  hy  Jj. ;  the  quotient  will  he  the  difference  of  longitude,  in  degrees, 
minutes,  and  seconds. 

£XAMPI.£S  FOR  PRACTICE. 

365.  1.  What  is  the  difference  in  the  longitude  of  New  York  and  San  Fran- 
cisco, their  difference  of  time  being  3  hr.  11  min.  56  sec. 

2.  The  longitude  of  Sitka  is  135°  18'  west.  What  is  the  longitude  of  the 
city  of  Jerusalem  if,  Avhen  it  is  9  o'clock  and  5  minutes  a.  m.  at  Sitka,  it  is  27 
minutes  and  4  seconds  after  8  P.  m.  in  Jerusalem? 

S.  Find  the  difference  in  latitude  of  Chicago,  situated  41°  54'  north,  and 
Valparaiso,  33°  4'  south. 


116  REDUCTION    OF   ENGLISH    MONEY. 

4.  What  is  the  latitude  of  Washington,  if  it  be  61°  46'  20'  north  of  Kio 
Janeiro,  and  the  latter  place  be  24°  54'  south  latitude? 

5.  When  it  is  20|  minutes  after  noon  at  Washington,  it  is  21  niin.  26  sec. 
before  noon  at  Chicago,  87°  30'  west.     What  is  the  longtitude  of  Washington  ? 

MISCELLANEOUS  EXAMPLiJS. 

366.  i.  A  messenger  leaves  the  Greenwich  Observatory,  westward  bound, 
at  noon,  Dec.  31,  and  by  a  uniform  rate  of  speed  encircles  the  globe  in  24  hours. 
Where  is  he  at  the  end  of  the  old  year? 

2.  Suppose  the  messenger  be  eastward  bound,  at  what  point  will  he  meet  the 
new  year? 

3.  When  it  is  20  minutes  past  10  a.  m.  at  Cape  Horn,  68°  west,  what  is  the 
time  at  Cape  of  Good  Hope,  18°  19'  east? 

4.  When  it  is  noon  at  London,  what  is  the  time  at  St.  Augustine,  81°  35' 
west?  At  Berlin,  13°  30' cast?  At  Xew  Orleans,  90°  west?  At  Sidney,  152°  20' 
east?     At  Paris,  20°  20'  22r  east?    At  Xew  York,  74°  3'  west? 


ENGLISH   MONEY. 

367.  EngHsh  or  Sterling  Money  is  the  legal  currency  of  Great  Britain. 

Table. 

4  farthings  (  far. )  =     1  penny d. 

12  pence  =      1  shilling s. 

20  shillings      =|l^;„3«"::r 

Scale,  ascending,  4,  12,  20;  descending,  20,  12,  4. 

368.  The  standard  unit  is  the  pound  sterling,  the  value  of  which,  in  United 
States  money,  is  shown,  together  with  the  other  coins,  in  the  following 

Comparative  Table. 


The  farthing  =  \%  cent. 


The  shilling  =  24^  cents. 


The  penny      =  2^  cents.  The  pound    =  14.8665. 

Remark. — The  farthing  is  but  little  \ised,  except  as  a  fractional  part  of  the  penny. 

COINS    OF    GREAT   BRITAIN. 

369.     The  gold  coins  are  tlie  sovereign  and  the  half-sovereign. 

Tte  silyer  coins  are  the  crown  (equal  to  5  shillings),  the  half-crown,  the 
florin  (equal  to  2  shillings),  the  shilling,  the  six-penny  and  three-penny  pieces. 

The  copper  coins  are  the  penny,  the  half-j^enny,  and  the  farthing. 

The  guinea  (equal  to  21  shillings)  and  the  half-guinea  are  in  xase,  but  are  no 
longer  coined. 


EEDUCTION   OF  ENGLISH   MONEY.  117 

REDUCTION   OF    ENGLISH    MONEY. 

370.  To  Reduce  English  Money  from  Lower  to  Higher  Denominations. 

Example. — Eeduce  13206  farthings  to  units  of  higiier  denominations. 

Operation.  Explanation.— Since  4  farthings  equal  one  penny,  13206  far- 

things equal  as  many  pence  as  4  is  contained  times  in  13206,  or 
4J_13306  far.  3301,  plus  2  remainder,  equal  3301  pence,  2  farthings;  since  13 

12  )  3301  d.  +  2   far.     pence  equal  1  shilling,  8801  pence  equal  275  shillings,  plus  1 
9nT"97"        -1-  1   r1  penny;  since  20  shillings  equal  1  pound,  275  shillings  equal  13 

_i-Jll^  ^'  +  -■■  ^-  pounds,  plus  15  shillings.     Therefore,  13206  farthings  equal  £13. 

£13.      +  15  S.         15s.  Id.  2 far. 
13206  far.  =  £  13,  15  s.  1  d.  2  far. 

ISillle.— Divide  by  the  units  iiv  the  scale  from  the  given  to  tlie  rcquived 
denomination . 

EXAMPLKS   FOR   PRACTICE. 

371.  Eeduce 

1.  5124  s.  to  pounds,  I  3.   13042  d.  to  pounds. 

2.  11916  far.  to  shillings.  I  4-   18T409  far.  to  higher  denominations. 

372.  To  Reduce  English  Money  from  Higher  to  Lower  Denominations. 
Example. — How  many  farthings  in  £9,  4  s.  3  d.  2  far.? 


Explanation.— Since  1  pound  equals  20  shillings,  9  pounds 
equal  180  shillings,  and  180  shillings,  plus  4  shillings,  equal  184 
shillings;  since  1  shilling  equals  12  pence,  184  shillings  equal  2208 
pence,  and  2208  pence,  plus  3  pence,  equal  2211  pence;  since  1 
penny  equals  4  farthings,  2211  pence  equal  8844  farthings,  and 
8844  farthings,  plus  2  farthings,  equal  8846  farthings.  Therefore, 
£9,  4  s.  3  d.  2  far.  =  8846  far. 


2211  d. 

Rule. — Multiply  by  the  units  in  the  scale  from  the  given  to  the  7'equired 
denomination. 

EXAMPLES   FOR  PRACTICE. 

373.  1.     How  many  pence. in  £27? 

2.  How  many  farthings  in  19  s.  11  d.  ? 

3.  How  many  pence  in  £161,  17  s.  9  d.  ? 

.4.     Reduce  £41,  1  s.  10  d.  2  far.  to  farthings. 
5.     How  many  farthings  in  £13,  15  s.  1  d.  2  far.  ? 

374.  To  Reduce  English  Money  to  Equivalents  in  United  States  Currency. 
Example.— Reduce  £15,  3  s.  7  d.  2  far.  to  dollars  and  cents. 

First  Explanation.— Since  £1  equals  $4.8665,  £15  equal  $72.9975;  since  1  shilling  equals 
24i  cents,  3  shillings  equal  $.73;  since  1  penny  equals  2/j  cents,  7  pence  equal  $.1414  ;  since 
1  farthing  equals  1%  cent,  2  farthings  equal  $.0101.  Therefore,  £15,  3  s,  7  d.  3  far.  = 
$73.8789,  or  $73.88. 


Operation. 

£  9,  4  s. 

3  d.  2  far. 

20 

Operation 

180  8. 

Continued 

4  s. 

2211  d. 

184  s. 

4 

12 

8844  far. 

2208  d. 

2  far. 

3d. 

8846. 

118  REDUCTION    OF    ENGLISH    MONEY. 

Second  Expl.vn-atiox. — Call  each  2  shillings  -j^  of  a  pound,  then  3  shillings  equal  £.15; 
call  the  pence  and  farthings,  reduced  to  farthings,  so  many  yg'j^g  of  a  pound,  then  7  pence,  plus 
2  farthings,  equal  30  farthings,  equal  £.030;  to  these  add  the  £15,  and  the  result  is  £15.18. 
And,  since  £1  equals  14.8665,  £15.18  equal  15.18  times  $4.8665,  or  $73.88,  as  before  found. 

Rules. — 1.  Multiply  each  of  the  orders  of  Sterling  Jtwiiey  by  its  equiva- 
lent iji   United  States  currency,  and  add  the  results.    Or, 

2.  Reduce  the  Sterling  expression  to  pounds  and  decimals  of  a  pound 
by  calling  each  2  shillings  to  of  a  pound,  and  the  pence  antl  farthings, 
reduced  to  faHhings,  so  inany  nsW  of  a  pound;  multiply  the  pounds  and 
decimals  of  a  pound  thus  obtained  by  Ji..8665,  and  the  product  will  be 
the  answer  in  dollars  and  cents. 

Remark. — This  is  exact  to  within  -s\^  of  the  part  represented  by  the  pence  and  farthings. 

KXAMPUi.S  FOR  PKACTICK. 

375.  Reduce  to  equivalents  in  United  States  money 

1.  £71,  19  s.  5  d.  and  3  far. 

2.  £108,  11  d.  and  1  far. 

3.  £13057,  10  s.  and  -i  d. 
J^.  £3,  1  s.  9  d.  and  2  far. 
5.  £11,  3  s.  Id.  1  far. 

376.  To  Reduce  "United  States  Money  to  Sterling  equivalents. 
Example. — Reduce  851G4.28  to  equivalents  in  English  money. 

Operation. 


4.8665  )  5104.28 


£1061  +  £ 

.189  rem. 

20 

3  s. 

+  .78  s. 
12 

rem, 

9d. 

+  36  d. 
4 

rem, 

Explanation.— Since  $4.8665  equal  £1,  $5164.28 
equal  £1061.189;  multiply  the  decimal  by  the  units 
in  the  scale,  20,  12,  4,  in  order,  pointing  off  as  in 
decimals,  and  obtain  3  s.  9  d.  1  far.,  which,  added  to 
the  £1061,  equals  £1061,  3  s.  9  d.  4  far. 


1  far.  +  .44  far. 


Rule. — Divide  the  expression  of  decimal  cujTency  by  ^.8665,  and  the 
integers  of  the  quotient  will  be  pounds  Sterling ;  reduce  the  decimal  of 
the  quotient,  if  any,  by  multiplying  by  the  loicer  units  in  tJie  scale. 

EXAMPLKS   rOK   PRACTICE. 

377.     1.     Reduce  $185  to  equivalents  in  English  money. 

2.  Reduce  $308.50  to  equivalents  in  English  money. 

3.  Reduce  $2500  to  equivalents  in  English  money. 

Jf.     Reduce  $3658.21  to  equivalents  in  English  money. 
6.     Reduce  $110085.75  to  equivalents  in  English  money. 


REDUCTION   OF    DENOMINATE   NUMBERS.  119 

MEASURES   OF   WEIGHT. 

378.  Weight  is  the  measure  of  gravity,  and  is  of  three  kinds,  distinguished 
from  each  other  by  their  uses,  viz  : 

Troy  loeight,  with  units  of  j^ounds,  ounces,  pennyweights,  and  grains,  used  for 
weighing  precious  metals. 

Avoirdupois  weight,  with  units  of  tons,  hundred  weights,  pounds,  ounces,  and 
drams,  used  for  weighing  products  and  general  merchandise. 

Apothecaries'  weight,  with  units  of  pounds,  ounces,  drams,  scruples,  and  grains, 
used  by  druggists. 

TROY   WEIGHT. 

379.  The  Troy  pound  is  the  standard  of  weight,  and  is  equal  to  2:^2.7944 
cubic  inches  of  pure  water,  at  its  greatest  density.  The  grains  of  the  other 
weights  are  the  same  as  the  Troy  grains. 

Table. 

24  grains  (gr.)    =  1  pennyweight pwt.  ^ 

20  pennyweiglits  =  1  ounce oz. 

12  ounces  =:  1  pound lb. 

Scale  -I  descending,  12,  20,  24.  |  1  lb.  =  5760  grains.      ~Z 

I  ascending,  24,  20,  12.  I  1  oz.  =    480  grains'  ^   ^ 


REDUCTION  OF  DENOMINATE  NUMBERS. 

380.     To  Reduce  Denominate  Numbers  from  Higher  to  Lower  Denominations. 

Example— Reduce  G  lb.  11  oz.  15  pwt.  21  gr.,  Troy,  to  grains. 

Operation.  ^                     yiust    Explanation.  —  Since   1   pound  equals  13 

6  lb.  11  oz.  15  pwt.  21  gr.       ounces,  6  pounds  equal  72  ounces,  and  72  ounces  plus 

J^  11  ounces  equal  83  ounces ;  since  1  ounce  equals  20 

72  oz.  pennyweights,  83  ounces  equal  1660  pennyweights,  and 

11  oz.  1660  pennyweights  plus  15  pennyweights  equal  1675 

"gg  Qj,  pennyweights;  since  1  pennyweight  equals  24  grains, 

2Q       '  1675  pennyweights  equal  40200  grains,  plus  21  grains 

equal  40221  grains.      Therefore,  6  lb.  11  oz.  15  pwt. 

1660  pwt.  21  gr.  Troy,  =  40221  gr. 
15  pwt. 

Second  Explanation.— Since  1  pound  equals  576C 

^,  .  1       '  grains,  6  pounds  equal  84500  grains;  since  1  ounce  equals 

480  grains,  11  ounces  equal  5280  grains  ;  since  1  pen- 

40200  gr.  nyweight  equals  24  grains,  15  pennyweights  equal  360 

21  gr.  grains;  to  these  add  the  21  grains,  and  the  entire  sum  is 

40221  gr.  ^2^1  S*"*'°^- 

Remark.— A  thorough  knowledge  of  the  unit  equivalents,  together  with  readiness  in  the 
use  of  the  multiplication  table,  renders  the  second  form  much  the  shorter  of  the  two  methods. 


120  REDUCTION    OF    DENOMINATE    NUMBERS. 

Rules. — 1.  Multiply  the  units  of  the  highest  denomination  given  bjf 
that  nurtibcr  in  the  scale  which  will  reduce  it  to  the  denomination 
next  lower,  and  add  the  units  of  that  lower  denomination;  continue 
in  this  manner  until  the  required  denomination  is  reached.    Or, 

2.  Multiply  the  units  of  each  denomination  by  the  nuTnber  of  units  of 
the  desired  equivalent  u'liich  it  takes  to  mahe  one  of  that  denomination, 
and  add  the  products  thus  obtained. 

381.  To  Reduce  Denominate  Numbers  from  Lower  to  Higher  Denominations. 

Example. — Reduce  40'^"^!  gr.,  Troy,  to  higher  denominations. 

First  Operation.  t?.-  a-        ^a        •  •,  -,  -i,* 

Explanation. — Smce  24  grains  equal  1  pennyweight, 

24  )  402^1  gr.  40221  grains  equal  1675  pennyweights,  plus  21  grains;  since 

•^0  'k  M"^~     wt  4-  21  o-r  ^^  pennyweights  equal  1  ounce,  1675  pennyweights  equal 

' '  83  ounces,  plus  15  pennyweights;  since  12  ounces  equal 

12  )  83  oz.     +  15  pwt.  1  pound,  83  ounces  equal  6  pounds,  plus  11  ounces.    There- 

,,  , ,  fore,  40221  gr.,  Troy,  =  6  lb.  11  oz.  15  pwt.  21  gr. 

6  lb.     +  11  oz.  ..  6    .        J  i' 

40221  gr.,  Troy,  =  6  lb.  11  oz.  15  i)wt.  21  gr. 

Second  Operation. 

5760  )  40221  gr.  (  6  lb.  Explanation.— Since  5760  grains  equal  1  pound,  40221 

34560  grains  equal  6  pounds,  plus  5661  grains;  since  480  grains 

dsTTV^fifiT  o-  •    M  1  o  equal  1  ounce,  5661  grains  equal  11  ounces,  plus  381  grains; 

5280                       '  since  24  grains  eqnal  1  pennyweight  381  grains  equal  15 

pennyweights,  plus  21  grains.    Therefore,  40221  gr.  =  6  lb. 

24  )  381  gr.  (  15  pwt.  n  oz.  15  pwt.  21  gr.,  as  before  found. 
360 

21  gr. 
40221  gr.,  Troy,  =  6  lb.  11  oz.  15  pwt.  21  gr. 

Remark. — The  first  form  is  advised  for  practice,  as  the  operations  may  usually  be  per- 
formed by  short  division. 

Example  2. — Reduce  11426  gr.,  Troy,  to  higher  denominations. 
Explanation. — Divide  the  given  number  by  24,  the  integers  of  the  quotient  by  20,  the 
integers  of  the  new  quotient  by  12. 

Rules. — 1.    Divide  by  the  successive  units  in  the  scale.    Or, 

2.    Divide  by  the  unit  equivalents  of  each  of  the  higher  denominations. 

382.  To  Reduce  Denominate  Fractions  from  a  Higher  to  a  Lower  Denomination, 
Example. — Reduce  ^^Vfr  l^^-?  Troy,  to  the  fraction  of  a  pennyweight. 

First  Operation. 
^^  X  V  X  V  =  ilU  =  If-         Explanation.— t^Vtj  of  a  PO"nd  equals  ^/^  of  the  12 

ounces  in  1  pound,  or  ^f^^  ounces;  ^fl^y  of  an  ounce 
Second  Operation.  ^^"^^"^  tUtt  of  the  20  pennyweights  in  1  ounce,  or  f|f§, 

4  which  equals  |f  pennyweights.     Therefore,  j/^^j  lb., 

T^  X  V  X  ^  =  M  pwt.    Troy,  =  |^  pwt. 

3  t 

"Rnie.— Multiply  the  fraction  hy  Ihe  units  in  the  scale,  from  the  given 
to  the  required  denomination. 


REDUCTION    OF    DENOMINATE    NUMBERS.  121 

383.  To  Reduce  a  Denominate  Fraction  from  a  Lower  to  a  Higher  Denomi- 
nation. 

Example.  —Reduce  |  of  a  grain  to  the  fraction  of  a  pound,  Troy. 

O  "PER  ATION 

Explanation.— I  of  a  grain  equals  f  of  ^  of  a  pen- 

fX^X^XiV^  Tjhnr-       nyweiglit ;  |  of  -^  of  a  pennyweight  equals  |  of  ^  of 

12  ^ff  of  ^^  ounce ;  f  of  ^  of  ^^  of  an  ounce  equals  f  of  ^ 

of  ^  of  yij  of  a  pound,  or  xrinTT  of  a  pound. 
*gr.  =  rrhirl^-'  Troy. 

Rule. — Divide  by  the  units  in  the  scale,  from  the  given  to  the  required 
denomination. 

384.  To  Reduce  Denominate  Fractions  to  Integers  of  Lower  Denominations. 

Example. — Reduce  y\  of  a  pound,  Troy,  to  integers  of  lower  denominations. 
Opekation. 

^  Explanation. — ^\  of  a  pound  equals  -^^  of  the  13  ounces 

•^  X  V       t  *^^'        '^t  ^'^-       in  a  pound,  or  \\  ounces,  which  reduced  gives  2^  ounces  ; 

*  i  of  an  ounce  equals  \  of  the  20  pennyweights  in  an  ounce, 

1   ^  ^ K        f  or  5  pennyweights.     Therefore,  y^j  of  a  pound,  Troy,  equals 

'  2  ounces,  5  pennyweights. 

^  lb.,  Troy,  ==  2  oz.  5  pwt. 

Rule. — Multiply  the  denominate  fraction  hy  the  unit  next  lower  in 
the  scale,  and  if  the  product  he  an  iinproper  fraction  reduce  it  to  a 
whole  or  mixed  number. 

385.  To  Reduce  a  Compound  Denominate  Number  to  a  Fraction  of  a  Higher 
Denomination. 

Example. — Reduce  7  oz.  5  pwt.  9  gr.  to  the  fraction  of  a  pound,  Troy. 

Operation.  First  Explanation. — Since  1  ounce  equals  20  pennyweights, 

~         -       ,j.    q      .  7  ounces  equal  140  pennyweights;  140  pennyweights  plus  5  pen- 

'      ^       '      °  *         ny weights  equals  145  pennyweights ;  since  1  pennyweight  equals 

24  grains  145  pennyweights  equal  8480  grains;  8480  grains  plus  9 

140  pwt.  grains  equals  3489  grains;  since  1  pound  equals  5760  grains,  3489 

5  pwt.  grains  equal  f  ^|^  of  a  pound. 

145  pwt.  Second  Explanation. — Since  1  ounce  equals  480  grains,   7 

OA  ounces  equal  83G0  grains;  since  1  pennyweight  equals  24  grains, 

-; 5  pennyweights  equal  120  grains;  3360  grains,  plus  120  grains,  plus 

o4:hU  gr.  9  grains  equal  3489  grains;  since  1  pound,  Troy,  equals  5760 grains, 

Q^gi"'  3489  grains  equal  ||§|  of  a  pound.     Therefore,  7  ounces,  5  pen- 

3489  ST.  =  ?f  g§  lb.  nyweights,  9  grains,  equal  |lf§  of  a  pound,  Troy. 

YivXe.— Reduce  the  compound  denominate  number  to  its  lowest  denomi- 
nation for  a  numerator,  and  a  unit  to  the  same  denomination  for  a 
denominator;  the  fraction  thus  formed  is  the  ansiver  sought. 


122  ADUITIOX    OF    DENOMINATE    NUMBERS. 

386.     To  Beduce  a  Denominate  Decimal  to  Units  of  Lower  Denominations. 

Example. — Reduce  .805  of  a  pound,  Troy,  to  integers  of  lower  denominations. 
Operation. 
.865  lb. 

1  .^  Explanation. — .865  of  a  pound  eqxials  .865 

^  of  the  12  ounces  in  1  pound,  or  10.38  ounces; 

10.380  oz.  .38of  an  ounce  equals.  38  of  the  20  penny  weights 

~  0  in  1  ounce,  or  7.6  pennyweights;  .6  of  a  penny- 

■^  60  pwt  weight  equals  .6  of  the  24  grains  in  1  penny- 

n  .  weight,  or  14.4  grains.     Therefore,  .865  of  a 

pound,  Troy,  equals  10  ounces,  7  pennyweights, 

14.4  gr.  14.4  grains. 
,865  lb.,  Troy,  =  10  oz.  7  pwt.  14.4  gr. 

Rule.— Multiply  tlie  decimal  by  tJiat  unit  in  the  scale  which  udll  reduce 
it  to  units  of  the  next  loxver  denomination,  and  in  the  product  point  off 
as  in   decimals.    Proceed  in  liJce  manner  with  all  decimal  remainders. 

3JS  7     To  Reduce  Denominate  Numbers  to  Decimals  of  a  Higher  Denomination. 
Example. — Reduce  8  oz.  3  pwt.  15  gr.  to  the  decimal  of  a  pound,  Troy. 
Operation. 

24 )  15  o"r 

1 °  ■  Explanation. — Since  24  grains  equal  1  pennyweight,  15 

.  625  grains  equal  V\  or .  625  of  a  pennyweight ;  3  pennj-weights  plus 

3.          pwt.  .62o  pennyweights  equal  3.625  pennyweights;  since  20  penny- 

20  ">  S  6^5  T)wt  weights  equal  1  ounce,  3.625  pennyweights  equal  .18125  of  an 

— '- *  ounce,  and  8  ounces  plus  .18125  of  an  ounce  equal  8.18125 

.18120  oz.  ounces;  since  12  ounces  equal  1  pound,  8.18125  ounces  equal 

8.              oz.  .68177iVof  apound.     Therefore,  8  oz.  3  pwt.  15  gr.=.68177i 

12  )  8.18125  oz.  It)'  Troy. 

.6817:tV1>J- 
YiViX^.— Divide  the   lowest  denomination  given  hy  the  number  in  tJie 
scale  next  higher,  and  to  the  quotient  add  the  integers  of  the  next  higher 
denomination .    So  continue  to  divide  by  all  the  successive  orders  of  units 
in  the  scale. 

ADDITION   OF    DENOMINATE   NUMBERS. 

388.  Example. — Find  the  sum  of  2  lb.  5  oz.  13  pwt.  4  gr.,  17  11).  11  oz. 
18  pwt.  20  gr.,  and  9  lb.  9  oz.  6  pwt.  15  gr. 

Explanation. — Since  each  of  the  given  expressions  is  a 
compound  number  of  the  same  class,  and  they  all  have  the 
same  varying  scale,  their  addition  may  be  performed  the  same 
as  in  simple  numbers;  in  reducing  the  sum  of  each  column 
from  a  lower  to  a  higher  order,  observe  the  units  in  the 
ascending  scale. 
30  lb.   2  oz.  18  pwt.  15  gr. 

Rule.— I  Write  the  nunibers  of  the  same  unit  value  in  the  same 
column . 

II  Beginning  with  the  lowest  denomination,  add  as  in  simple  numbers, 
and  reduce  to  higher  denominations  according  to  the  scale. 


Operation. 

lb. 

oz. 

pict. 

P-- 

2 

5 

13 

4 

17 

11 

18 

20 

9 

9 

6 

15 

Ih. 

oz. 

pwf. 

gr. 

23 

4 

17 

6 

11 

1 

13 

y 

11  lb. 

9oz 

.     3  pwt. 

21  gr 

MULTIPLICATION   OF   DENOMINATE   NUMBERS.  123 

SUBTRACTION  OF  DENOMINATE  NUMBERS. 

389.     Example.— Subtract  11  lb.  7  oz.  13  pwt.  9  gr.  from  ^3  lb.  -i  oz.  17  pwt. 

6  gr. 

Explanation. — Subtract  as  in  simple  numbers.  If  a 
subtrahend  term  be  numerically  greater  than  the  cor- 
responding minuend  term,  borrow  1  from  the  next  higher 
minuend  term,  reduce  it  to  equivalent  units  in  the  denom- 
ination next  lower,  add  them  to  the  minuend  units,  and  from 
their  sum  take  the  subtrahend  units. 

Yi\\\e.— Write  the  numbers  as  for  simple  subtraction ;  take  each  subtra- 
hend term  from  its  corresponding  minuend  term  for  a  remainder.  In 
case  any  subtrahend  term  he  greater  than  the  minuend  term,  borrow  1 
as  in  simple  subtraction,  and  reduce  it  to  the  denomination  required. 


MULTIPLICATION   OF    DENOMINATE    NUMBERS. 

300.  Example. — Each  of  five  bars  of  silver  weighed  IG  lb.  3  oz.  10  pwt.  21 
gr.     What  was  the  total  weight? 

Explanation. — Multiply  21  grains  by  5  and  obtain  105 

Operation.  grains,  which  reduce  to  pennyweights  by  dividing  by  24, 

21  Q2        yii^i        Qj-  and  obtain  4  pennyweights,  with  a  remainder  of  9  grains; 

,p  q  -irv  f)-|  multiply  10  pennyweights  by  5,  add  the  4  pennyweights, 

and  reduce  to  ounces  bj'  dividing  by  20,  obtaining  2  ounces, 

iL.  14  pennyweights;  multiply  3  ounces  by  5,  add  the  2  ounces 

81  lb.    5  oz.   14  pwt.  9  gl".  and  divide  by  12,  obtaining  1  pound,  5  ounces;  multiply  16 

pounds  by  5,  add  the  1  pound  and  obtain  81  pounds. 

Rule. —  Beginning  ivith  the  lowest  denomination,  multiply  each  in 
succession,  and  reduce  the  product  to  higher  denominations  by  the  scale. 

Remarks. — 1.  In  order  that  the  pupil  may  have  all  problems  under  each  denominate  subject 
given  together,  and  so  make  an  exhaustive  study  separately  of  each,  it  has  seemed  proper 
to  include  all  of  the  reductions  under  a  typical  subject,  that  of  Troy  Weight,  and  hereafter, 
as  may  be  needed,  reference  will  be  made  to  such  reductions. 

2.  The  teacher  will  appreciate  the  above  change,  as  each  subject  will  thus  be  made  to 
include  enough  work  for  a  lesson,  and  the  confusion  often  arising  from  giving  in  the  same 
lesson  several  tables,  with  varying  scales,  may  be  avoided. 


DIVISION   OF   DENOMINATE   NUMBERS. 

391.     Example.— If  7  lb.  7  oz.  12  pwt.  18  gr.  of  silver  be  made  into  G  i)late8 
of  equal  weight,  what  will  be  the  weight  of  each? 

Operation.  Explanation. — One  plate  will  weigh  J  as  much  as 

lb.        oz.      Vict.  nr.  Opiates.     Write  the  dividend  and  divisor  as  in  short  divi- 

g-vi*-  ^  1.)  -.o  sion.     Divide  7  pounds  by  6,  obtaining  a  quotient  of  1 

; ;^ pound  and  an  undivided  remainder  of  1  pound;  reduce 

1  lb.    3  oz.     5  i)wt.  1 1  gr.         this  remainder  to  ounces (12)  and  add  to  the  7  ounces  of 
the  dividend,  obtaining  19  ounces,  which  divide  by  6,  obtaining  3  ounces  and  an  undivided 


124  COMPOUND    DEXOMINATE    DIVISION. 

remainder  of  1  ounce;  reduce  this  remainder  to  pennyweights  (20)  and  add  to  the  12  penny- 
weights of  the  dividend,  obtaining  32  pennyweights,  which  divide  by  6,  obtaining  5  penny- 
weights and  an  undivided  remainder  of  2  pennyweights;  reduce  this  remainder  to  grains  (48) 
and  add  to  the  18  grains  of  the  dividend,  obtaining  66  grains,  which  divide  by  6,  obtaining  11 
grains,  and  thus  completing  the  division.  Therefore,  the  weight  of  each  plate  will  be  1  pound, 
3  ounces,  5  pennyweights,  11  grains. 

Unle.— Write  the  terms  as  in  short  division;  divide  as  in  integers, 
and  reduce  remainders,  if  any,  to  next  lower  orders  hy  the  scale. 

Remarks. — 1.  Should  the  highest  dividend  order  not  contain  the  divisor,  reduce  its  units 
to  the  order  next  lower,  and  so  proceed  to  the  end. 

2.  The  above  and  like  divisions  may  be  accomplished  by  the  reduction  of  the  denominate 
expressions  to  the  lowest  order  in  its  scale,  then  effecting  the  division  and  afterwards  reducing 
the  quotient  to  higher  denominations. 


COMPOUND   DENOMINATE    DIVISION. 

392.  Example. — How  many  plates,  each  weighing  1  lb.  3  oz.  5  pwt.  11  gr., 
can  be  made  from  7  lb.  7  oz.  12  pwt.  18  gr.  of  silver? 

Explanation. — Reduce  each  of  the  given 

Operation.  expressions  to  its  equivalent  in  grains.     Since 

1  lb.  3  oz.  5  pwt.  11  gr.  =  7331  gr.  one  plate  weighs  7331  grains,  and  the  weight  of 

-.!,'»      '  -.cT       ',     ,o    *  Anr^^^'  the  silver  to  be  used  is  43986  grains,  as  many 

7  lb.  7  oz.  12  pwt.  18  gr.  =  43986  gr.        ,  ,  ,         ,        ^,        •  t.   ^  i  / 

r  to  to  plates  can  be  made  as  the  weight  of  one  plate, 

43986  gr.  ^  7331  gr.  =  6  7331  grains,  is  contained   times  in  the  43986 

grains  to  be  so  used,  or  6  plates. 

Rule. — Reduce  the  dividend  and  divisor  to  the  same  denomination, 
and  divide  as  in  simple  numbers. 

JKXAMPLKS   FOK   PRACTICE. 

393.  1.     Reduce  31  lb.  10  oz.  13  jiwt.  to  pennyweights. 
2.     How  many  grains  in  27  lb.  17  pwt.  20  gr.  ? 

S.  How  many  pounds,  ounces,  and  pennyweights  in  230.51  gr.  .^ 

J^.  Reduce  30297  grains  to  higher  denominations. 

5.  Reduce  -^^  of  a  pound  to  grains. 

6.  ^tVtt  of  a  pound  is  what  part  of  a  pennyweight  ? 

7.  -^7  of  a  grain  is  what  fraction  of  an  ounce? 

8.  Reduce  -^  of  a  pennyweight  to  the  fraction  of  a  i)()und. 

9.  Reduce  ^  of  a  pound  to  lower  denominations 

10.  Reduce  f  of  an  ounce  to  lower  denominations. 

11.  Reduce  9  oz.  1  i)wt.  21  gr.  to  the  fraction  of  a  pound. 

12.  What  fraction  of  a  pound  equals  11  oz.  11  pwt.  18  gr.  ? 

13.  What  is  the  value  in  lower  denominations  of  .6425  lb.  ? 
H.  Find  the  equivalents  in  lower  denominations  of  .905  oz.  ? 

15.  3  oz.  11  pwt.  12  gr.  is  what  decimal  of  a  pound  ? 

16.  Reduce  17  pwt.  12  gr.  to  the  decimal  of  an  ounce. 

17.  Add  236  lb.  4  oz.  15  pwt.,   83  lb.  11  oz.  21  gr.,   4()  11>.  l»i  pwt..    l(»o  lb. 
9  oz.  II  gr. 


AVOIRDUPOIS   WEIGHT.  125 

18.  What  is  the  sum  of  16  lb.  16  pwt.  16  gr.,  100  lb.  1  oz.  5  pwt.  20  gr., 
76  lb.  7  oz.  6  pwt.  13  gr.,  19  lb.  2  oz.  10  pwt.  20  gr.? 

19.  Find  the  equivalents  in  lower,  denominations  of  .1425  oz. 

20.  1  pwt.  15  gr.  is  what  decimal  of  a  pound  ? 

21.  Subtract  41  lb.  11  oz.  6  pwt.  18  gr.  from  50  lb.  2  oz. 

22.  What  is  tlie  difference  between  19  lb.  9  oz.  11  pwt.  and  11  oz.  16  pwt.  22  gr.  ? 
2S.     What  will  be  th6  cost  of  15  gold  chains,  each  weighing  1  lb.  3  oz.  18  pwt. 

18  gr.,  at  7^  per  grain  ? 

2J^.  I  bought  7  lb.  7  oz.  12  pwt.  18  gr.  of  old  gold,  at  81.05  j^er  pwt.  AVhat 
was  the  sum  paid  ? 

25.  A  manufacturer  made  18  vases  from  7  lb.  8  oz.  8  pwt.  18  gr.  of  silver. 
What  was  their  average  weight  ? 

26.  If  12  rings  be  made  from  1  lb.  8  oz.  of  gold,  what  will  be  the  weight 
of  each  ? 

27.  A  miner  having  77  lb.  10  oz.  5  pwt.  of  gold  dust,  divided  \  of  it  among 
nis  laborers,  and  had  the  remainder  made  into  chains  averaging  3  oz.  3  pwt.  3 
gr.  of  pure  gold  each.  If  he  sold  the  chains  for  $52.50  each,  how  much  did  he 
receive  for  them  ? 

28.  What  is  the  aggregate  weight  of  five  purchases  of  old  silver,  weighing 
respectively  4  lb.  9  oz.  20  gr.,  13  lb.  17  pwt.  22  gr.,  20  lb.  1  oz.  17  pwt.  4  gr., 
8  lb.  2  oz.,  and  27  lb.  12  pwt.  21  gr.? 

29.  I  bought  27  lb.  11  oz.  1  gr.  of  old  silver,  and  after  having  used  15  lb.  15 
pwt.  15  gr.,  sold  the  remainder  at  5^  per  pwt.  What  quantity  was  sold,  and  how 
much  was  received  for  it  ? 

80.  A  goldsmith  bought  3  lbs.  9  oz.  1  pwt.  16  gr.  of  old  gold,  at  80^  per  pwt., 
and  made  it  into  pins  of  40  grains  weight  each,  which  he  sold  at  $2  apiece. 
How  much  did  he  gain  or  lose  ? 

AVOIRDUPOIS   WEIGHT. 
394-.     Avoirdupois  Weight  is  used  for  all  ordinary  purposes  of  weighing. 

Table. 

16  ounces =  1  pound lb. 

100  pounds z=  1  hundred-weight. .  cwt. 

20  hundred-weight.,  or  2000  pounds  =  1  ton T. 

Scale,  descending,  20,  100,  16  ;   ascending,  16,  100,  20. 

Remark. — At  the  United  States  Custom  Houses,  in  weighinggoods  on  which  duties  are  paid 
and  to  a  limited  extent  in  coal  and  iron  mines,  the  long  ton  of  2240  pounds  is  still  used. 

Long  Ton  Table, 

16  ounces =  1  pound lb. 

28  pounds =  1  quarter qr. 

4  quarters,  or  112  pounds =  1  hundred-weight...  cwt. 

20  hundred-weight,  or  2240  pounds  =  1  ton T. 


126 


AVOIRDUPOIS  WEIGHT. 
Table  of  Avoirdupois  Pounds  per  Bushel. 


Wojth.  Territory. 

0     X     X        ' 

CD     W     ©»         1 

0 
0 

CO    0    0    » 

»     0     CD      0 

i  § 

Wisconsin 

00 

>* 

0    X    X    «o 
0    cj    cj    la 

CO 
0 

C» 

eo 

0      CD 
CO      O 

CO      0 

•*    CO 

Vertnont 

CO 
IS 

CJ 

0      CO 

CO       LO 

0 

CO 

Bhode  Island.  .. 

; 

; 

;  g  g    : 

i 

Pennsylvania.  .. 

0 

X 

0 

g 

'■•   CO 
■      ts 

s 

Oregon 

01 

->* 

0    X    00 

0     (M     <M 

CO 
to 

S 

0    CO 

CD     0 

0 

CO 

Ohio.. 

3D 

; 

0 
CD 

0 
0 

CO 
10 

§? 

'   ^ 

0 
0 

New  York 

X 

§ 

IS 

§ 

00 

0     CO     CO     -^     0 

CO    ia    i-o    ■^    CD 

New  Jersey. 

X 

0 
IS 

0 

0 

CO 
0 

0 

eo 

0    0 
CO    0 

0 
CO 

New  Hampshire. 

X 

0 

-<* 

; 

; 

0 

CD 

North  Carolina. 

' 

; 

; 

0 

0 

CO 

1 

Missouri 

xoO'*'««0'<J<=reo-<*« 

!-•?    I-    0    CO    0    >.":    0    0 

sc     0     CO     13     0     -*     CO     CJ 

Minnesota. 

X 

ci 

■^ 

0     X     X 
CS>     OJ      « 

CO 

10 

CJ 

so 

CO 

0 

0 

CD 

Michigan 

X 

5? 

0     X     X 

0      CJ      CI 

CO 
us 

CJ 

00 

CO 

0 

CO 

Massachusetts.  .. 

^ 

CO 
0 

eo    ic 

» 
0 

0 
CO 

Maine. 

■ 

0 
eo 

0 

CO 

I 

Louisiana. 

0 

CJ 

eo 

; 

0 
CO 

Kentucky.  

X    o    o    -*    -?> 
-r    o     X     —    1-t 

0 

»     -*     «D 
0     -^     IC 

eo    0    CO    o    Lo    tJI    CO    c* 

Iowa 

xoo-fc-»oo-^:c«o-*»XL-t-05DOooo 
■^;cx^is-*off*eoo-*io;oe30i5i.';i0'*0(?» 

Indiana 

xoO'rooo««eo»-^ox»>xc^o»-'5o       > 
'T    o    e-    i-i    o    -*<    «    ci    «    »o    -r    L-;    ;c    ec    'T    ec    Lt    «    Tf    ee 

Illinois - 

LO       0      0 

'*      CO      CJ 

Connecticut ; 

1    -i^ 

; 

CO 
10 

X 

S    «5 

5        ' 

California 

o 

i   0 

CJ 
1.0 

so 

-1' 

0 

CO 

CO 

o 

O 

■>    a 

C 

c 

,      C 

.   C 

'     a 

'     C 

,    c 

:  1 

'  CO 

)     S     es 

d     & 

5    5    ^ 

-     C 
cS 
C 

c: 

u 

"5 

JS 

(■ 

a 

'     > 

.    c 

-   _a 

< 

•    c 

a 

: 
5    P 

> 

cc 

B 

i. 

-    c 

_    c 

a 

c 

a 

.1  . 

21 

a 
■I 

c 

a 
C 

1 

> 

1 

> 

J 

5 

--     ^-^    '■:     v:     f-    '-c     c^     ::     •-    '^,    '^-     -t-    i;;     ;:^     ?^    v;     cv     ^    *^    ^    05 

EXAMPLES   FOR    PRACTICE. 


127 


Additional  Table  of  Weights  of  Products, 

As  usually  given,  but  varied  by  tbe  laws  of  different  States  : 


Apples,  green, 56  lb.  per  bushel. 

Charcoal,  . .  - .22  lb.  per  bushel. 

Hungarian  Grass  Seed, 45  lb.  per  bushel. 

Malt, 38  lb.  per  bushel. 

Millet, .45  lb.  per  bushel. 


Mineral  Coal, 80  lb.  per  bushel. 

Peas, 60  lb.  per  bushel. 

Potatoes,  sweet, 55  lb.  i)er  bushel. 

Red  Top  Grass  Seed, . .  14  lb.  per  bushel. 
Turnips, .56  lb.  per  bushel. 


Table  of  Gross  Weights  for  Freighting. 


Ale  and  Beer, .330  1b.  per  barrel. 

Apples, 150  lb.  per  barrel- 
Beef  (200  lb.  net),. 330  lb.  per  barrel. 

Cider, .400  lb.  per  barrel. 

Corn  Meal, .200  lb.  per  barrel. 

Eggs, 180  lb.  per  barrel. 

Fish, 300  lb.  per  barrel. 

Flour  ( 196  lb.  net ),  200  lb.  per  barrel. 


High  wines, 400  lb.  per  barrel. 

Lime, 230  lb.  per  barrel. 

Oil, 400  lb.  per  barrel. 

Pork  (200  lb.  net), ,330  lb.  per  barrel. 

Potatoes, 180  lb.  per  barrel. 

Salt, ...300  1b.  per  barrel. 

Vinegar, 400  lb.  per  barrel. 

Whiskey, 400  lb.  per  barrel. 


Estimates  on  Lumber,  Wood,  Etc.,  foi*  Freighting. 

Pine,  Hemlock,  and  Poplar,  seasoned,  per  M, 3000  lb. 

Black  AValnut,  Ash,  Maple,  and  Cherry,  per  M, . . .  4000  lb. 

Oak  and  Hickory,  per  M, _ 5000  lb. 

Soft  wood,  dry,  per  cord, — 3000  lb. 

Hard  wood,  dry,  per  cord, 3500  lb. 

Remark. — For  unseasoned  lumber,  add  one-third. 


Brick,  common,  each, 4  lb. 

Brick,  fire,  each, 6  lb. 


Sand,  cubic  yard, 3000  lb. 

Gravel,  cubic  yard, 3200  lb. 


Stone,  cubic  yard, 4000  lb. 

Remark. — For  assistance  in  the  solution  of  the  following  examples,  the  pupil  is  referred  to 
the  explanations  and  rules  under  Troy  Weight. 


KXAMPLES   FOR  PRACTICE. 

395.     i.     How  many  pounds  Avoirdupois  in  17  T.  6  cwt.  69  Ih.  'i 
^      Reduce  31275  lb.  Avoirdupois  to  higlier  denominations, 
f  of  a  ton  Avoirdupois  equals  how  many  pounds  ? 
Reduce  -^-^  of  a  cwt.  Avoirdupois  to  ounces. 
Reduce  .3842  of  a  ton  Avoirdupois  to  lower  denominations. 
How  many  Avoirdupois  pounds  in  .625  of  a  ton  ? 
17  cwt.  72  lb.  4  oz.  Avoirdupois  is  what  fraction  of  a  ton  ? 
Reduce  51  lb.  12  oz.  Avoirdupois  to  the  fraction  of  a  hundred- weight. 
What  decimal  part  of  a  hundred-weight  is  24  lb.  2  oz.  Avoirdupois  ? 
Reduce  19  cwt.  99  lb.  15  oz.  Avoirdupois  to  the  decimal  of  a  ton. 


3. 

4. 

5 . 

6. 

7. 

8. 
■  9. 
JO. 


128  apothecaries'  weight. 

11.  Wliat  is  the  sum  of  T  T.  4  cwt.  78  lb.  5  oz.,  3  T.  17  cwt.  19  lb.  11  oz., 
5  T.  18  cwt.  96  lb.,  13  T.  1  cwt.  11  oz.  ? 

12.  A  farmer  sold  -4  loads  of  hay,  weighing  respectiyely  1  T.  2  cwt.  14  lb., 
19  cwt.  90  lb.,  1  T.  3  cwt.  97  lb.,  1  T.  5  cwt.,  and  received  for  it  $16  per  ton. 
How  much  did  he  receive? 

13.  Six  loads  of  lime  weighed  13  T.  15  cwt.  4  lb.  Wliat  was  their  average 
weight? 

APOTHECARIES'   WEIGHT. 

396.  Apothecaries'  Weight  is  used  by  druggists  in  retailing,  and  by 
apothecaries  iu  mixing  medicines. 

Table. 

20  grains      =  1  scruple sc. 

3  scruples  =  1  dram dr. 

8  drams      =  1  ounce  .  - oz. 

12  ounces     =  1  pound lb. 

Scale,  descending,  12,  8,  3,  20  ;  ascending,  20,  3,  8,  12. 

Remarks. — 1.     The  pound,  ounce,  and  grain  are  the  same  as  in  Troy  weight.     The  only 
difference  between  these  weights  is  in  the  subdivisions  of  the  ounce. 
2.     Drugs  and  medicines  are  sold  at  wholesale  by  Avoirdupois  weight. 

EXAMPLES  FOK  PRACTICE. 

Remark. — For  assistance,  refer  to  rules  and  explanations  under  Troy  "Weight. 

397.  1.     Eeduce  5128  sc.  to  higher  denominations. 

2.  How  many  drams  in  61  lb.  5  oz.  ? 

3.  10  oz.  1  dr.  1  sc.  15  gr.  equal  what  fraction  of  a  pound  ? 
J^.     Eeduce  .955  of  a  pound  to  lower  denominations. 

5.  How  many  scruples  in  -^  of  a  pound  ?  ' 

6.  Add  6/o  lb.,  7y5^  oz.,  3|  dr.  and  2|  sc. 

7.  Find  the  sum  of  ^-i  lb.,  7  oz.,  7  dr.,  1  sc.  and  16  gr. 

8.  From  21  lb.  5  oz.  3  dr.  1  sc.  11  gr.,  take  14  lb.  1  oz.  7  dr.  19  gr. 

9.  What  is  the  difference  between  16  lb.  1  oz.  4  dr.  2  sc.  12  gr.,  and  '^^^  lb.? 

10.  In  comjjounding  six  cases  of  medicine,  an  apothecary  used  for  each  2  lb. 
7  oz.  6  dr.  18  gr.     What  was  the  aggregate  weight  ? 

11.  If  19  lb.  4  oz.  7  dr.  1  sc.  5  gr.  be  divided  into  21  packages  of  equal  weight, 
what  will  be  the  weight  of  each  ? 

Comparative  Table  of  Weights. 

Troy.  Apothecaries.'  Avoirdupois. 

1  pound  =  5760  grains  =  5760  grains  =  7000      grains. 
1  ounce  =    480  grains  =    480  grains  =    437.5  grains. 
175  pounds  =     175  pounds  =    144       pounds. 
QuESTioxs. — 1.     Which  is  heavier,  a  pound  Troy  or  a  pound  Avoirdupois  ? 
2.     Which  is  heavier,  an  ounce  Troy  or  an  ounce  Avoirdupois  ? 


MEASURES   OF   CAPACITT.  129 

Remarks.— 1.     A  cubic  foot  of  water  weighs  62^  lb.  or  1000  oz.  Avoirdupois. 

2.  In  weighing  diamonds  and  gems,  the  unit  generally  employed  is  the  carat,  which  is 
«bout3.3  Troy  grains. 

3.  The  term  carat  is  also  used  to  express  the  fineness  of  gold,  24  carats  fine  being  pure; 
thus  18  carat  gold  =  f  pure. 

EXAMPLES  FOR  PKACTICE. 

398.  1.  A  dealer  bought  131  lb.  5  oz.  of  drugs  by  Avoirdupois  Aveight,  at 
t;G.25  per  pound,  and  retailed  them  at  5^  per  scruple.     What  was  his  gain  ? 

2.  How  much  is  gained  or  lost  by  buying  23  lb.  4  oz.  of  medicine  by  Avoirdu- 
pois weight,  at  50^  per  oz.,  and  selling  by  Apothecaries  weight,  at  1^^  per  grain? 

S.     Reduce  5f  lb..  Avoirdupois,  to  Troy  units. 
•     Jf..     What  is  the  remainder  after  subtracting  hl^^  lb.  Troy  from  60  lb.  10  oz. 
Avoirdupois  ? 

5.  I  bought  by  Avoirdupois  weight  45  lb.  6  oz.  of  drugs,  and  from  the  stock 
sold  by  Apothecaries  weight  29  lb.  4  oz.  3  dr.  1  sc.  10  gr.  What  is  the  remainder 
worth,  at  75j^  per  Avoirdupois  ounce  ? 

6.  Having  bought  \l  of  a  pound  of  roots  by  Avoirdupois  weight,  I  sold  H  of 
a  pound  by  Apothecaries  weight.  What  was  the  remainder  worth,  at  10^  per 
scruple  ? 


MEASURES   OF   CAPACITY. 

390.  Dry  Measure  is  used  for  measuring  grains,  seeds,  fruit,  vegetables, 
•etc. — all  articles  not  liquid. 

'J'ho  units  are  pints,  quarts,  pecks,  and  bushels. 

Table. 

2  pints  (pt.)  =  1  quart qt. 

8  quarts  =  1  peck. pk. 

4  pecks  =  1  bushel bu. 

Scale,  descending,  4,  8,  2  ;  ascending,  2,  8,  4. 

Remakks. — 1.  The  United  States  Standard  Unit  of  Dry  Measure  is  the  bushel,  which,  as  a 
•circular  measure,  is  18i  inches  in  diameter  and  8  inches  deep,  contains  2150.42  cubic  inches, 
and  is  in  uniform  use  for  measuring  shelled  grains;  while  the  Jieaped  bushel  of  2747.71  cubic 
inches  is  used  for  measuring  apples,  roots,  and  corn  unshelled. 

2.  The  British  Imperial  bushel  contains  2318.19  cubic  inches.  The  English  Quarter  men- 
tioned in  prices  current  =  8  bu.  of  70  lb.  each,  or  560  lb.  avoirdupois  =  \  long  ton. 

3.  For  weights  of  different  commodities,  refer  to  Table,  page  125. 

EXAMPLES  EOK    PKACTICE. 

Remark. — For  assistance,  refer  to  rules  under  Tkoy  Weight. 

400.     1.     How  many  pints  in  14^  bu.  ? 
2.     Reduce  9  bu.  1  pk.  3  qt.  1  pt.  to  i)ints. 

S.     Add  51  bu.  3  \)k.  1  pt. ;  4G  bu.  2  pk. ;  37  bu.  2  (it.  1  pt. ;  51  bu.  1  pk.  7  qt. 
4-     From  I  of  a  bushel,  take  f  of  a  peck. 
9 


130  LIQUID   MEASURE. 

5.  What  is  the  difference  between  Tt  bu.  and  2  bu.  2  pk.  2  qt,  1  pt.  ? 

6.  A  teamster's  12  loads  of  wheat  measured  1000  bu,  1  pk.  6  qt.  1  pt.     How 
much  was  the  average  of  each  load  ? 

7.  What  will  be  the  cost,  at  45^  per  bushel,  of  5  loads  of  oats,  weighings 
respectively  2619  lb.,  2554  lb.,  2124  lb.,  3051  lb.,  and  2745  lb.? 


LIQUID   MEASURE. 

401.  Liquid  Measure  is  used  for  measuring  water,  oil,  milk,  cider,, 
molasses,  etc. 

The  units  are  gills,  pints,  quarts,  gallons,  and  barrels. 

Table. 

4  gills  (gi.)  =  1  pint. pt. 

2  pints  =  1  quart qt. 

4  quarts        =  1  gallon gal. 

31^  gallons  =  1  barrel bar.  or  bbl. 

Scale,  descending,  31^,  4,  2,  4 ;  ascending,  4,  2,  4,  31^. 

Remarks. — 1.  The  standard  unit  of  Liquid  Measure  is  the  gallon,  which  contains  231 
cubic  inches.  , 

2.  Casks,  called  hogsheads,  pipes,  butts,  tierces,  tuns,  etc.,  are  indefinite  measures;  their 
capacity,  being  determined  by  gauging,  is  usually  marked  upon  them. 

3.  In  sales  of  oils  and  liquors,  and  in  certain  other  cases,  the  barrel  is  also  of  indefinite 
capacity. 

EXAMPUES  FOR    PKACTICE. 

Remark. — For  assistance  refer  to  rules  under  Troy  Weight. 

402.  1.     How  many  gills  in  5  bar.  27  gal.  3  qt.  1  pt.  of  cider? 

2.  Reduce  31479  gi.  to  higher  denominations. 

3.  .046  of  a  barrel  equals  how  many  gills? 

4.  From  .895  of  a  barrel  take  21  gal.  2  qt.  1  pt.  1  gi. 

5.  From  a  cask  containing  68  gal.  4^  qt.  of  wine,  1.625  bar.  were  sold.  What 
was  the  remainder  worth,  at  50^  per  pint  ? 

6.  A  reservoir  contained  896  gal.  2  qt.  of  water,  and  its  contents  were  put 
into  116  kegs.     What  was  the  quantity  put  into  each  ? 

7.  From  |  of  a  barrel  take  4  gal.  1  qt.  3  gi. 

8.  If  2  qt.  1  pt.  1  gi.  of  oil  be  consumed  per  day  for  the  year  1890,  what  will 
be  its  cost  for  the  year  at  8^  per  gallon  ? 

9.  From  a  cask  of  brandy  containing  69  gal.  1  pt.  and  costing  $3.75  per 
gallon,  one-fourth  leaked  out,  and  the  remainder  was  sold  at  20^  per  gill.  What 
was  the  amount  of  gain  or  loss  ? 

Comparative  Table  of  Dry  and  Liquid  Measures. 

Cu.  in.  in  one  Cu.  in.  in  one  Cu.  in.  in  one 

gallon.  quart.  pint. 

Dry  Measure (^pk.)268f  67^  33|. 

Liquid  Measure 231  57|  28|. 


LINEAR   MEASTJBE.  131 

Remarks. — 1.  A  pint  of  water  weighs  about  1  pound,  Avoirdupois. 

2.  Potatoes  and  grains  are  tisually  sold  to  dealers  and  shippers  by  weight. 

3.  The  beer  gallon  of  282  cubic  inches  is  nearly  obsolete. 

EXAMPLES  FOR  PKACTICE. 

403.     1.     Keduce  21  bu.  6  qt.  1  pt.,  dry  measure,  to  pints,  liquid  measure. 

2.  A  grocer  bought  12  bu.  3  pk.  3  qt.  of  chestnuts  by  dry  measure  and 
when  selling  used  a  liquid  pint,  measure.  How  many  pints  did  he  gain  by  the 
change  ? 

3.  A  bushel  of  cherries,  bought  at  10^-  per  quart,  dry  measure,  was  sold  at 
the  same  jmce  per  quart,  wine  measure.     How  much  was  thereby  gained  ? 

4.  A  cask  of  cranberries,  containing  5^^  bu.,  was  bought  for  $15,  and  retailed 
at  10^  per  quart  by  wine  measure.     What  was  the  gain  ? 

5.  A  blundering  clerk  bought  of  a  gardener  375  quarts  of  currants,  measur- 
ing them  by  a  liquid  quart  measure,  and  when  selling  used  a  dry  quart  measure. 
If  he  bought  at  6^  per  quart  and  sold  at  7^,  how  much  less  did  he  receive  than  if 
he  had  measured  by  dry  measure  when  buying  and  by  liquid  measure  when 
selling  ? 


MEASURES   OF  EXTENSION. 

404.  Extension  is  that  which  has  one  or  more  of  the  dimensions,  length, 
breadth,  and  thickness;  it  may  therefore  be  a  line,  a  surface,  or  a  solid. 

405.  A  Line  has  only  one  dimension — length. 

Remakks.— 1.  The  United  States  Standard  of  linear,  surface,  and  solid  measure,  is  the  yard 
of  3  feet,  or  36  inches. 

2.  The  standard,  prescribed  at  Washington,  has  been  fixed  with  the  greatest  precision.  It 
was  determined  by  a  brass  rod,  or  pendulum,  which  vibrates  secomUin  a  vacuum  at  the  sea  level, 
in  the  latitude  of  London,  Eng.,  and  in  a  temperature  of  62°  Fahrenheit.  This  pendulum  is 
divided  into  391393  equal  parts,  and  360000  of  these  parts  constitute  a  yard. 

406.  A  Surface  or  Area  has  two  di\men&ion&— length  and  breadth. 

407.  A  Solid  has  three  diimen&ions— length,  breadth,  and  thickness. 

LINEAR    MEASURE. 

408.  Linear  or  Long  Measure  is  used  in  measuring  lengths  and  distances. 

Table. 

12  inches  (in.) =  i  foot ft. 

3  feet =1  yard yd. 

5^  yards,  or  1G|  feet  =  1  rod rd. 

320  rods.. =  1  statute  mile mi. 

Scale,  descending,  320,  5^  3,  12  ;  ascending,  12,  3,  5^,  320. 
1  Mile  =  320  rods,  or  5280  feet,  or  63360  inches. 


132 


SQUARE    MEASURE, 


Special  Table. 


^  of  an  inch  =  1  Size,  applied  to  ^hoes. 
18  inches  =  1  Cubit. 
3.3  feet  =  1  Pace. 

5  paces  =  1  Rod, 

4  inches  =  1    Hand,    used    to  measure 
the  height  of  animals. 

6  feet  =  1    Fathom,  used    to    measure 
depths  at  sea. 

1.152|  statute  miles  =  1  Geographic  or 
Nautical   mile,   used   for   measuring 


3  geographic  miles  =  1  League,  used  for 
measuring  distances  at  sea. 

00  geographic  miles  or  69.16  statute 
miles  =  1  Degree  of  Latitude  on  a 
meridian,  or  Longitude  on  the  equa- 
tor. 

360  degrees  =  Equatorial  circumfer- 
ence of  the  earth. 

1  geographic  mile  =  1  Knot,  used  to 
determine  the  speed  of  vessels. 


distances  at  sea. 

Remarks. — 1.  In  civil  engineering,  and  at  the  Custom  Houses,  the  foot  and  inch  are  divided 
into  tenths,  hundredths,  and  thousandths. 

2.  The  yard  is  divided  into  halves,  quarters,  eighths,  and  sixteenths,  for  measuring  goods 
sold  by  the  yard. 

3.  The  furlong  of  40  rods  is  little  used. 

4.  Deirrees  are  of  unequal  length;  those  of  latitude  varying  from  68.72  miles  at  the  Equator 
to  69.3-1  miles  in  the  polar  regions.  The  average  length,  69.16  miles,  is  the  standard  adopted 
by  the  United  States  Coast  Survey. 

5.  A  degree  of  longitude  is  69.16  statute  miles  at  the  equator,  but  decreases  gradually  toward 
the  poles,  where  it  is  0. 

KXA3IPI.ES   FOK   PRACTICE. 

Remark. — For  assistance  refer  to  Rules  under  Troy  Weight. 

409.  1.     Eeduce  2  mi.  1  rd.  7  ft.  to  inches. 
Reduce  2501877  inches  to  higher  denominations. 
AVhat  part  of  a  mile  is  ^V  of  a  foot  ? 
Reduce  f  of  a  mile  to  integers  of  lower  denominations. 
What  fraction  of  a  rod  is  11  ft.  2  in.  ? 
Reduce  .542  of  a  mile  to  integers  of  low^er  denominations. 
Reduce  285  rd.  7  ft.  4  in.  to  the  decimal  of  a  mile. 
A  wheelman  ran  71  mi.  246  rd.  1  yd.  2  ft.  6  in.  in  the  forenoon,  and  20 

mi.    10    rd.   8   in.    less  in  the   afternoon.     What  distance  did  he  run    in    the 
entire  day  ? 

9.  If  a  yacht  makes  an  average  of  227  mi.  227  rd.  2  yd.  2  ft.  2  in.  per 
day,  for  the  seven  days  of  a  week,  what  distance  will  be  passed  ? 

10.  If  the  Sei'via  steams  2905  mi.  in  six  days,  what  is  her  average  rate  per  day? 

SQUARE  MEASURE. 

410.  Square  Measure  is  used  for  computing  the  surface  of  land,  floors, 
boards,  walls,  roofs,  etc. 

411.  The  Area  of  a  figure  is  the  quantity  of  surface  it  contains. 

412.  An  Angle  is  the  difference  in  the  direc- 
tion of  two  lines  j)roceeding  from  a  common  point 
called  the  vertex. 


2. 
3. 

4. 
o. 
6. 

7. 
8. 


Angle. 


SQUARE    MEASURE. 


133 


Two  Right  Angles. 


413.  A  Right  Angle  is  the  angle  formed 
when  one  straight  line  meets  another  so  as  to 
make  the  adjacent  angles  equal.  The  lines  form- 
ing the  angles  are  said  to  he  2)erpe7idicidar  to  each 
other.  E  and  F  are  right  angles,  and  the  lines 
A  B  and  C  D  are  perpendicular  to  each  other. 


Rectangle. 


3  inches. 


Contents 


Six 


inclies. 


3X2  in.  =6  8q.  in. 
3  feet. 

'•one!  '  ! 

^°^r   [QUE 


square: 


YARD 


414.  A  Rectangle  is  a  plane  or  flat  surface, 
having  four  straight  sides  and  four  square  corners, 
or  four  right  angles. 


415.  The  Contents  or  Area  of  any  surface 
having  a  uniform  length  and  a  uniform  breadth  is 
found  by  multiplying  the  length  by  the  breadth. 
In  the  accompanying  diagram, in  which  the  angles 
{a,  b,  c,  d),  are  all  right  angles,  and  the  corners  all 
square  corners,  the  area  is  6  square  inches,  and  is 
found  by  multiplying  2  inches  by  3  inches. 


416.  A  Square  is  a  figure  bounded  by  four  equal 
lines,  and  having  four  right  angles. 

Remark. — A  square  inch  is  a  square,  each  side  of  which  is 
1  inch.  A  square  foot  is  a  square,  each  side  of  which  is  1  foot. 
A  square  yard  is  a  square,  each  side  of  which  is  1  yard. 


3  X  3  ft.  =  9  sq.  ft.  =  1  sq.  yd. 


Table  of  Square  Measure. 

144  square  inches  (sq.  in.) =  1  square  foot. 

9  square  feet =  1  square  yard . 


sq.  ft. 
sq.  yd. 


30i  square  yards,  or  (                       _.                   ,  j 

272i  square  feet  ....\    -  -  -  -  i  square  loci sq.  i a. 

IGO  square  rods =  1  acre A. 

040  acres =  I  square  mile sq.  mi. 

36  square  miles  (6  miles  square),  =  1  township Tp. 

Scale,  descending,  36,  640,  160,  30^,  9, 144;  ascending,  144,  9,  30^,  160,  640,  36. 

Remark. — All  the  units  of  square  measure,  except  the  acre,  are  derived  by  squaring  the 
corresponding  units  of  linear  measure;  as,  a  square  foot  is  a  surface  one  foot  square;  a 
square  rod  is  a  surface  1  rod  or  16^  feet  square;  a  square  mile  is  a  surface  1  mile  or  320  rods 
square. 


134  SQUARE  MBASFEE. 

417.  The  Unit  of  Land  Measure  is  the  acre,  equal  to  208.71ft.  x 
208. 71  ft. 

Remarks. — 1.  In  sections  of  the  United  States  where  the  original  grants  were  from  France, 
the  arpent,  a  French  unit  of  surface,  equal  to  about  %  of  an  acre,  is  still  sometimes  used. 
2.  The  Rood,  equal  to  40  square  rods,  is  but  little  used. 

418.  Dimension  stuff  is  sold  by  hoard  measure. 

419.  The  Unit  of  Board  Pleasure  is  a  square  foot  surface,  oue  inch 
thick,  called  a  hoard  foot. 

420.  To  Find  the  Number  of  Board  Feet  in  a  Board. 

llule. — Multiply  the  length  in  feet  hy  the  ividth  in  iivches,  and  divide 
by  12;  the  quotient  jrill  he  the  iiumher  of  square  feet. 

Rkmark. — If  the 'board  tapers  evenly,  find  the  mean  or  average  width,  by  adding  the 
width  of  the  two  ends,  and  dividing  by  2. 

4*21.     To  Find  the  Number  of  Board  Feet  in  Timbers  or  Planks. 

^w\e.— Multiply  the  length  in  feet  hy  the  product  of  the  ividth  and 
thickness  in  incites,  and  divide  by  12. 

422.  To  Find  the  Number  of  Squares  in  a  Floor  or  Roof. 

Remark. — In  flooring,  roofing,  slating,  etc.,  the  square,  or  100  square  feet,  is  used  as  a  unit 
of  measure. 

Rule.— Poi;/^  off  two  decimal  places  from  the  right  of  the  numher  of 
surface  feet- 

423.  To  Find  the  Number  of  Yards  of  Carpeting  that  Would  be  Required  to 
Cover  a  Floor. 

Rule.— I.  Divide  one  of  the  dimensions  of  the  floor  by  3,  add  the 
wastage,  if  any,  and  the  result  ivill  be  the  length,  in  yards,  of  1  strip  of 
the  carpet. 

II-  Divide  the  other  dimension  by  tlie  width  of  the  carpet,  and  the 
quotient  will  be  the  iiuviher  of  strips  it  will  take  to  cover  the  floor. 

III.  Multiply  the  length  of  each  strip  by  the  number  of  strips,  and  the 
product  will  be  the  nmnhcr  of  yards  required. 

Remark. — In  carpeting  and  papering,  it  is  usually  necessary  to  allow  for  certain  waste  in 
matching  the  figures  of  patterns,  and  often  carpets  may  be  laid  with  less  waste  one  way  of  the 
room  than  the  other.     Dealers  charge  for  all  goods  furnished,  regardless  of  the  waste. 

KXAMPLKS   rOK   I'KACTICK. 

Remark. — For  assistance  refer  to  rules  under  Troy  Weight. 

424.  1.     Reduce  5  A.  110  sq.  rd.  7  sq.  ft.  to  square  inches. 
~.     Eeduce  4  sq.  mi.  527  A.  105^  S(|.  rd.  to  square  feet. 

3.     Reduce  .1754  of  a  S(iuare  mile  to  lower  denominations. 


SQUARE   MEASURE.  136 

4..     Reduce  \^  of  an  acre  to  lower  denominations. 

5.  What  fraction  of  a  square  mile  is  j\  of  a  square  foot? 

6.  "What   decimal    part  of   an  acre  is  150  sq.  rd.    3  sq.  yd.  7  scj.    ft.   100 


rsq.  in. 


V 


7.  From  .0375  of  an  acre  take  \^  of  a  square  rod. 

8.  To  the  sum  of  ^,  f,  and  -^  of  an  acre,  add  .0055  of  a  square  mile. 

9.  How  many  squares  in  a  roof,  each  side  of  which  is  2G  x  CO  feet? 

10.  How  many  yards  of  carpet,  1  yard  wide,  Avill  be  required  to  cover  a  floor 
10.5  yd.  long  by  6  yd.  Avide,  if  no  allowance  be  made  for  matching  ? 

11.  IIow  many  feet  in  8  boards,  each  15  ft.  long,  9  in.  wide,  and  1  in.  thick? 

12.  How  many  feet  in  15  boards,  each  IG  ft.  long  and  1  in.  thick,  the  boards 
being  13  in.  wide  at  one  end  and  10  in.  at  the  other? 

13.  How  many  acres  in  a  square  field,  each  side  of  which  is  04  rods  in 
length? 

14.  "What  will  be  the  cost  of  a  tract  of  land  508  rd.  long  and  1350  rd.  wide, 
at  $25  per  acre? 

13.  A  field  87^  rd.  wide  and  240  rd.  long,  produced  27f  bu.  of  wheat  to  the 
4icre.     What  Avas  the  crop  Avorth,  at  90^  per  bushel? 

16.  A  farm  in  the  form  of  a  rectangle  is  75  rd.  Avide;  if  the  area  is  107.5  A., 
hoAV  long  is  the  farm? 

17.  I  wish  to  build  a  shed  which  will  coA'er  f  of  an  acre  of  land.  If  the  Avidth 
of  the  shed  is  42  ft.,  what  must  be  its  length? 

15.  17.75  bu.  of  timothy  seed  is  sown  on  land,  at  the  rate  of  6  lb.  per  acre. 
What  is  the  area  thus  seeded? 

10.     What  is  the  difference  between  a  square  rod  and  a  rod  square? 

20.  What  is  the  difference  between  two  square  rods  and  tAvo  rods  square? 

21.  A  square  yard  will  make  how  many  surfaces  5  in.  by  9  in.  ? 

22.  IIow  many  acres  of  flooring  in  a  six-story  block  100  ft.  by  220  ft.  ? 

23.  A  rectangular  field  containing  10^  A.  is  45  rd.  wide.     What  is  its  length? 

24.  How  many  fields,  each  of  10  A.  50  sq.  rd.  21  sq.  yd.  5  sq.  ft.  and  28  sq. 
in.,  can  be  formed  from  a  farm  containing  124  A.  40  sq.  rd.  10  sq.  yd.  8  sq.  ft.  48 
sq.  in.  ? 

25.  HoAv  many  acres  in  v^  road  17200  ft.  long  and  00  ft.  wide? 

26.  AVhat  Avill  be  the  cost,  at  $3.50  per  M,  of  the  shingles  for  a  roof  26  ft. 
Avide  and  110  ft.  long,  if  the  shingles  are  0  in.  Avide  and  4  inches  of  their  length 
be  exposed  to  the  Aveather? 

27.  A  hall  7|  ft.  Avide  and  19|  ft.  long  is  covered  witli  oil  cloth,  at  05^  per 
.sq.  yd.     HoAv  much  did  it  cost? 

28.  If  a  farm  of  100  A.  94|  sq.  rd.  is  divided  equally  into  11  fields,  Avhat 
will  be  the  area  of  eacii  field  ? 

29.  Reduce  240089740  sq.  in.  to  higher  denominations. 

30.  HoAv  many  rods  of  fence  Avill  enclose  100  A.  of  land  lying  in  the  form 
of  a  square  ? 

31.  IIow  many  additional  rods  aviII  divide  the  farm  into  four  fields  of  ecjual 
.iirea  ? 


136  SC^lARE    MEASURE. 

32.  How  many  yards  of  brussels  carpeting,  f  of  a  yard  wide,  laid  length- 
wise of  the  room,  will  be  required  to  cover  a  room  23  ft.  by  17  ft.  4  in.,  if  the 
waste  in  matching  be    6  in.  on  each  strip  ? 

Remark. — When  the  width  of  the  room  is  not  exactly  divisible  by  the  width  of  the  carpet, 
drop  the  fraction  in  the  quotient  and  add  1  to  the  whole  number.     The  waste  in  such  cases  is. 
either  cut  off  or  turned  under  in  laying. 

3S.  AVhat  will  it  cost,  at  21^-  per  sq.  yd.,  to  plaster  the  sides  and  ceiling  of  a 
room  24  ft.  by  3U  ft.  and  10^  ft.  his:li,  if  one-sixth  of  the  surface  of  the  sides 
is  taken  up  by  doors  and  windows  ? 

34..  A  street  4975  ft.  long  and  40  ft,  wide  was  paved  with  Trinidad  asphaltum, 
at  $2. 65  per  square  yard.     What  was  the  cost  ? 

35.  A  skating  rink,  204  ft.  by  196^  ft.,  was  floored  with  2  in.  plank,  at  $23.50 
per  M.     What  was  the  cost  of  the  lumber  ? 

36.  What  will  be  the  cost  of  the  carpet  border  for  a  room  10^  ft.  by  21  ft.,  if 
the  price  be  G2^^'  per  yard  ? 

37.  How  many  single  rolls  of  paper,  8  yd,  long  and  18  in.  wide,  will  it  take  to 
cover  the  ceiling  of  a  room  56  ft.  long  and  27  ft.  4>in.  wide,  if  there  be  no  Avaste 
in  matching  ? 

Remark.— When  no  allowance  is  made  for  waste  in  matching,  divide  the  surface  to  be 
papered  by  the  number  of  square  feet  in  one  roll  of  the  paper. 

38.  How  many  yards  of  carpeting,  £  of  a  yard  wide,  Avill  be  required  to  carpet 
a  room  32  ft.  long  and  25  ft.  wide,  if  the  lengths  of  carpet  are  laid  crosswise  of 
the  room,  and  8  inches  is  lost  on  each  length  in  matching  the  pattern  ?  How 
many  yards  if  the  lengths  are  hiid  lengthwise  and  6  in,  is  lost  in  matching  ?  If 
the  carpet  is  laid  in  the  most  economical  way,  what  will  be  the  cost,  at  $2.55  per 
yard  ? 

39.  How  many  sheets  of  tin,  20  in.  liy  14  in.,  will  be  required  to  cover  a  roof 
60,5  ft,  wide  and  156.25  ft.  long  ? 

40.  What  is  the  difference  between  four  square  feet  and  four  feet  square  ? 

41.  What  will  it  cost,  at  $1.15  per  yard,  to  carpet  a  flight  of  stairs  11  ft.  4  in_ 
high,  the  tread  of  each  stair  being  10  in.  and  the  riser  8  in.? 

42.  How  many  shingles,  averaging  4  in,  wide  and  laid  5  in.  to  the  weather, 
will  cover  the  roof  of  a  barn,  one  side  of  the  roof  being  24  ft.  wide  and  the  other 
42  ft,  wide,  the  length  of  the  barn  being  60  ft.  ? 

43.  Divide  an  acre  of  land  into  8  equal  sized  lots,  the  length  of  each  of 
which  shall  be  twice  its  frontage.     What  will  be  the  dimensions  of  each  lot  ? 

44-  How  many  granite  blocks,  12  in.  by  18  in.,  Avill  be  required  to  pave  a  mile 
of  roadway  42  ft.  in  width  ? 

45.  What  will  be  the  coot,  at  20/'  per  s(|.  yd.,  for  plastering  the  ceiling  and 
walls  of  a  room  22  ft.  wide,  65  ft.  long,  and  15  ft.  high,  allowance  being  made 
for  8  doors  4  ft.  6  in.  wide  by  11  ft.  6- in.  high,  and  10  windows  each  42  in.  wide 
by  8  ft.  high  '' 

46.  I  wish  to  floor  and  ceil  a  room  27^^  yd.  long  and  15  yd.  2  ft.  wide,  with 
matched  pine.     What  will  be  the  cost  of  the  material,  at  |!26.40  i>er  M  ? 


SQUARE   ROOT.  137 

INVOLUTION. 

425.  A  Power  of  a  number  is  the  product  arising  from  multiplying  a 
number  by  itself,  or  repeating  it  several  times  as  a  factor. 

426.  A  Perfect  Power  is  a  number  that  can  be  exactly  produced  by  the 
involution  of  some  number  as  a  root;  thus,  G-i  and  16  are  perfect  powers,  because 
8x8  =64,  and  2  X  2  X  2  X  2  =  16. 

427.  The  Square  of  a  number  is  its  second  poiver. 

428.  The  Cube  of  a  number  is  its  third  power. 

429.  Involution  is  the  process  of  finding  any  power  of  a  number;  and  a 
number  is  said  to  be  involved  or  raised  to  a  power,  when  any  power  of  it  is  found. 

•  KXAMPLKS   rOIt   PKACTICE. 


430.     1'     What  is  the  square  of  1  ? 

2.  What  is  the  square  of  3  ? 

3.  What  is  the  square  of  4  ? 
Jf.     What  is  the  square  of  5  ? 


5.  What  is  the  square  of  9  ? 

6.  AVhat  is  tlie  square  of  10  ? 

7.  What  is  the  square  of  99  ? 

8.  What  is  the  square  of  250  ? 


Remark. — From  the  solution  of  the  above  examples  the  pupil  will  observe: 
1st.  That  the  square  of  any  number  expressed  by  one  figure  cannot  contain  less  than  1  nor 
more  than  2  places. 

2d.  That  the  addition  of  I  place  to  any  number  will  add  2  places  to  its  square. 


EVOLUTION. 

431.  Evolution  is  the  process  of  extracting  the  root  of  a  number  considered 
as  a  power.  It  is  the  reverse  of  Involution,  and  each  may  be  proved  by  the 
other. 

432.  A  Root  of  a  number  is  one  of  the  equal  factors  which,  multiplied 
together,  will  produce  tlie  given  number;  as,  4  x  4  x  4  =  64;  4  is  the  root  fnmi 
which  the  number  64  is  produced. 


SQUARE   ROOT. 

433.  The  Square  Root  of  a  Number  is  such  a  number  as,  multij)lied  by 
itself,  will  produce  the  required  number. 

434.  The  operation  of  finding  one  of  the  two  equal  factors  of  a  square,  or 
product,  is  called  extractinfj  the  square  root. 

Remark.— The  square  root  of  any  number,  then,  is  one  of  its  two  equal  factors,  the  given 
number  being  considered  a  product. 

435.  In  practical  operations,  a  surface  and  one  of  its  diuieusious  being  given, 
the  wanting  dimension  is  found  by  dividing  the  surface  ])y  tlie  given  dimension. 


138 


SQUARE   ROOT. 


The  accompanying  diagram  is  a  square  14  feet  by 
14  foot.  Its  square  feet,  or  area,  is  by  inspection 
found  to  be  made  up  of: 

1st.  The  tens  of  14,  the  number  representing  the 
length  of  one  side,  or  10  squared  —  100  square  feet, 
as  shown  by  the  square  within  the  angles  a,  b,  c,  d. 

2d.  Two  times  the  product  of  the  tc7is  by  the  nnits 
of  the  same  number,  or  2  (10  x  4)  =  80  square  feet, 
as  shown  by  the  surfaces  within  the  angles  e,  f,  g,  h, 
and  /,  j,  k,  I. 

3d.  The  square  of  the  units,  4  feet,  or  the  product 

of  4  ft.  by  4  ft.  =  16  square  feet,  as  shown  by  the 

square  within  the  angles  w,  x,  y,  z. 
14  Et.MO  Ft.  &  +  F  b.  ^ 

Hence,  a  square  14  feet  on  each  side  will  contain  10  x  10     =  100  square  feet. 

2  (10  X  4)  =    80  square  feet. 
4x4         =16  square  feet. 

196  square  feet. 
Or,  the  square  of  14  is  made  up  of  or  equals  the  square  of  10,  plus  twice  the  product  of  10 
by  4.  plus  the  square  4,  the  number  to  be  squared. 

436.  General  Priuciples. — Tlie  square  of  any  mimher  composed  of  tivo  or 
more  fif/ures  is  equal  to  the  square  of  the  tens,  plus  twice  the  inoduct  of  the 
tens  multiplied  hi/  the  units,  plus  the  square  of  the  iniits. 


a 

dl 

e 

h 

.A* 
h 

* 

oK 

L. 

b 

c 

f 

G 

II. 

i 

I 

w 

S 

^ 

J 

K 

X 

Y 

437 

I'.vrrs, 
1'  = 
2'  — 
3'  = 
4'  = 
5'  = 
6'  = 

»vj     

t     — 

b'  = 

9'  = 
10'  = 


TJnits  and  Squares  Compared. 
Sqiakks.         Remark. — Squaring  the  numbers  from  1  to  10  inclusive,  shows: 

1st.  That  the  square  of  any  number  will  contain  at  least  one  place,  or 
one  order  of  units. 

2d.  That  the  square  of  no  number  represented  by  a  single  figure  will 
contain  more  than  two  places.  If  the  number  of  which  the  square  root  is 
sought  be  separated  into  periods  of  two  figures  each,  beginning  at  the 
right,  the  number  of  periods  and  partial  periods  so  made  will  represent 
the  number  of  unit  orders  in  the  root.  Therefore,  the  square  of  any  num- 
ber will  contain  twice  as  many  places,  or  one  less  than  twice  as  many,  as 
its  root. 

3d.  "VThere  the  product  of  the  left  hand  figure  multiplied  by  itself  is  not 
greater  than  9,  then  the  square  will  contain  one  less  than  twice  as  many 
places  as  the  root. 


1 

4 

9 

16 

25 

36 

49 

64 

81 

100 


438.     Example. — Find  the  square  root  of  625. 


Operation. 


0.25)2  5 
4=  400 


ExPL.\NATioN. — The  number  consists  of  one  full  and 
one  partial  period;  hence  its  root  will  contain  ^?ro  places 
— tens  and  units.  The  given  number,  G25,  must  be  the 
product  of  the  root  to  be  extracted  multiplied  by  itself; 
therefore,  the  first  figure  of  the  root,  which  will  be  the 
highest  order  of  units  in  that  root,  must  be  obtained 
from  the  first  left  hand  period,  or  highest  order  of  units 
in  the  given  number.  Hence,  the  first  or  tens  figure  of 
the  root  will  be  the  square  root  of  the  greatest  perfect  square  in  6.  6  coming  between  4,  the 
square  of  2,  and  9,  the  square  of  3,  its  root  must  be  2  tens  with  a  remainder.     Subtracting 


20  X  2  =  40 
5 

45 


225 
225 


rem. 


0   rem. 


SQUARE   ROOT.  139 

from  the  6  hundreds  or  6,  the  square  of  2  (tens)  =  400  or  4,  gives  225  as  a  remainder.  Having 
now  taken  away  the  square  of  the  tens,  the  remainder,  225,  must  be  equal  to  2  times  the  tens 
multiplied  by  the  square  of  the  units,  plus  the  square  of  the  units.  Since  the  tens  are  2  or  20, 
twice  the  tens  =  40.  Observe,  therefore,  that  225  must  equal  40  times  the  zinits  of  the  root, 
together  with  the  square  of  such  units.  If,  then,  225  be  divided  by  40,  the  quotient,  5,  will 
nearly,  if  not  exactly  represent  the  units  of  the  root  sought.  Using  40,  then ,  as  a  trial  divisor,  the 
second,  or  unit  figure  of  the  root  is  found  to  be  5.  The  term,  ticice  th€  tens  multiplied  by  the 
units,  is  equal  to  2  (20  X  5),  or  200,  and  the  units,  or  5,  squared  =  25;  the  sum  of  these  wanting 
terms,  or  225,  is  the  remainder,  or  what  is  left  after  taking  from  the  power  the  square  of  the 
first  figure  of  the  root.     Therefore,  the  square  root  of  625  is  25. 

Rule.— I-  Beginning  at  the  right,  separate  the  given  niunher  into 
periods  of  two  places  each. 

II.  Take  the  square  root  of  the  greatest  perfect  square  contained  in  the 
left  hand  period  for  the  first  root  figure ;  subtract  its  square  from  the 
left  hand  period,  and  to  the  remaUvder  hring  down  the  next  period. 

III.  Divide  the  Tiumher  thus  obtained,  exclusive  of  its  units,  by  twice 
the  root  figure  already  found  for  a  second  quotient,  or  root  figure;  place 
this  figure  at  the  right  of  the  root  figure  before  found,  and  also  at  the 
right  of  the  divisor;  multiply  the  divisor  thus  formed  by  the  new  root 
figure,  subtract  the  result  from  the  dividend,  and  to  the  remainder  bring 
down  the  next  period,  and  so  proceed  till  the  last  period  has  been  brought 
down,  considering  the  entire  root  already  found  as  so  many  tens,  in 
deteimining  subsequent  root  figures- 

Rem.\rks. — 1.  Whenever  a  divisor  is  greater  than  the  dividend,  place  a  cipher  in  the  root 
and  also  at  the  right  of  the  divisor;  bring  down  another  period  and  proceed  as  before. 

2.  When  the  root  of  a  mixed  decimal  is  required,  form  the  periods  from  the  decimal  point 
right  and  left,  and  if  necessary  supply  a  decimal  cipher  to  make  the  decimal  periods  of  two 
places  each. 

3.  A  root  may  be  carried  to  any  number  of  decimal  places  by  the  use  of  decimal  periods. 

4.  Any  root  of  a  common  fraction  may  be  obtained  by  extracting  the  root  of  the  numerator 
for  a  numerator  of  the  root,  and  the  root  of  the  denominator  for  the  denominator  of  the  root. 

5.  To  find  a  root,  decimally  expressed,  of  any  common  fraction,  reduce  such  common  frac- 
tion to  a  decimal,  and  extract  the  root  to  any  number  of  places. 


KXAMPLESi  I'OK   VKACTICK, 

4:31).     1.     Find  the  square  root  of  ]  96. 

2.  Find  the  square  root  of  225. 

S.  Find  the  square  root  of  144. 

Jf.  Find  the  square  root  of  576. 

5.  Find  the  square  root  of  1225. 

6.  Find  the  square  root  of  5025. 

7.  Find  the  square  root  of  42436. 

8.  Find  the  square  root  of  125.44. 

9.  Find  the  square  root  of  50.2681. 

10.  Find  tlie  square  root  of  482,  carried  to  three  decimal  places. 

11.  Find  the  square  root  of  25.8,  carried  to  two  decimal  places. 


140  SQUARE   ROOT. 

12.  Find  tlie  square  root  of  106.413,  carried  to  four  decimal  places. 

13.  What  is  the  square  root  of  -j^  ? 
IJf.     What  is  the  square  root  of  ff  ? 

15.  What  is  the  square  root,  decimally  expressed,  of  ||,  carried  to  three 
decimal  places  ? 

-76\  What  is  the  square  root,  decimally  expressed,  of  ^\\,  carried  to  two 
decimal  places? 

11.     What  is  the  square  root  of  30368921,  carried  to  one  decimal  place. 

18.     What  is  the  square  root  of  4698920043,  carried  to  two  decimal  places. 

4:40.     A  Triangle  is  a  plane   figure  having  three 

sides  and  three  angles. 

4-il.     The  Base  is  the  side  on  which  the  triangle 
stands;  as,  a,  c. 

442.  The    Perpendicular   is  the  side   forming  a 
right  angle  with  the  base;  as,  a,  h,  in  fig.  S. 

443.  The   Hypothenuse   is  the   side  opposite  the 

TRIANGLE.  ^.jgj^^  ^^gj^.     ^g^    ^^^    ^^    jj^   ^g_    g_ 

Fig.  T.  is  a  triangle,  having  angles  at  a,  h,  c. 
Fig.  S. 

444.  A  Right-angled  Triangle  is  a  triangle  liaving  a 
right  angle. 

Fig.  S  is  a  right-angled  triangle,  the  angle  at  b  being  a  right 
angle.  The  line  a  h  is  the  Perpendicular;  tlie  line  h  c  \s  the 
Base;  the  line  ac  is  tlie  Hypothenuse. 

Remark. — It  is  a  geometrical  conclusion  that  the  square  formed  on  the 
hypothenuse  is  equal  to  the  sum  of  the  squares  formed  on  the  base  and 
the  peqiendicular 


RIOHT-ANOLEIJ 
TRIANGLE. 


445.     To  find  tlie  liy])Otlieiiuse,  when  the  base  and  perpendicular  are  given. 

Rule. — To  the  square  of  the  base  add  tlie  square  of  tlie  iierpendicular,  and 
extract  the  square  root  of  their  sum. 

To  find  the  base,  when  the  hypothenuse  and  perpendicular  are  given. 

Rule. — From  the  square  of  the  h^jpothcnuse  take  the  square  of  the  perpen- 
dicular, and  extract  the  square  root  of  the  remainder. 

To  find  the  perpendicular,  when  the  hypothenuse  and  base  are  given. 

Rule. — Take  the  square  of  the  base  from  the  square  of  the  hypothenuse,  and 
extract  the  square  root  of  tlie  remainder. 

i:XAJ»IPL,K8  FOK   PRACTICE. 

446.  1.  The  base  of  a  figure  is  G  ft.  and  the  perpendicular  8  ft.  Find 
the  hypothenuse. 

2.  The  perpendicular  is  17.5  ft.  and  the  base  is  46.6  ft.  Find  the  hypoth- 
enuse to  three  decimal  places. 


SURVEYOR  S  LONG  MEASURE.  141 

S.     The  hypothenuse  is  110  ft.  and  the  base  is  19.5  ft.     Find  the  perpendic- 
ular to  two  decimal  places. 

Jf..  Tlie  hypothenuse  is  86  ft.  and  the  base  is  equal  to  the  perpendicular.  Find 
both  of  the  wanting  terms  to  two  decimal  places. 

5.  The  hypothenuse  is  127  ft.  and  the  base  is  equal  to  ^  of  the  perpendicular. 
Find  wanting  terms  to  three  decimal  places. 

Remarks. — 1.  Observe,  in  example  4,  that  the  square  root  of  l_  the  square  of  the  hypothenuse 
is  equal  to  the  base;  and  in  example  5,  that  the  square  root  of  \  of  the  square  of  the  hypothenuse 
is  equal  to  the  base. 

2.  Carry  all  roots  to  two  decimal  places. 

6.  "What  is  the  length  of  one  side  of  a  square  field,  the  area  of  which  is  one 
acre  ? 

7.  How  many  feet  of  fence  will  enclose  a  square  field  containing  five  acres? 
8    I  wish  to  lay  out  ten  acres  in  tlie  form  of  u  square.     What  must  be  its 

frontage  in  feet  and  inches? 

9.     What  is  the  distance  from  the  top  of  a  perpendicular  flag-staff  105  ft. 
high  to  a  point  4  rods  from  the  base  and  on  a  level. with  it? 

10.  What  is  the  width  of  a  street  in  which  a  ladder  60  ft.  long  can  be  so 
placed  that  it  will  reach  the  eaves  of  a  building  40  ft.  high  on  one  side  of  the 
street,  and  of  another  building  50  ft.  high  on  the  opposite  side  of  the  street? 

11.  What  length  of  line  will  reach  from  the  lower  corner  to  the  opposite 
upper  corner  of  a  room  64  ft.  long,  27  ft.  wide,  and  21  ft.  high? 

12.  If  a  farm  be  one  mile  square,  how  far  is  it  diagonally  across  from  corner 
to  corner?     Find  the  answer  in  rods,  feet,  and  inches. 

13.  IIow  many  rods  of  fence  will  enclose  a  square  field  containing  20  acres? 
14-     A  farm  of  180  acres  is  in  the  form  of  a  rectangle,  the  length  of  which  is 

twice  its  width.     How  many  rods  of  fence  will  enclose  it? 

15.  AVhat  will  be  the  base  line  of  a  farm  of  136  A.  40  sq.  rd.  if  it  is  in  the 
form  of  a  right-angled  triangle,  with  the  base  equal  to  the  perpendicular? 


SURVEYOR'S   LONG    MEASURE. 

447.     The  Unit  of  measure  used  by  land  surveyors  is  Gunter's  Chain,  4  rods, 
or  QQ  feet,  in  length,  and  consisting  of  100  links. 

Remark. — Rods  are  seldom  used  in  Surveyor's  Measure,  it  being  customary  to  give  distances 
jn  chains  and  links  or  hundreths. 

Table. 

7.92  inches =  1  link  ...  1. 

25  links =1  rod rd. 

4  rods,  or  66  feet . . .  =  1  chain  . .   ch. 

80  chains,  or  320  rods  =  1  mile  . . .  mi. 

Scale,  descending,  80,  4,  25,  7.92;  ascending,  7.92,  25,  4,  80. 


142 


SURVEYOR  S   SQUARE   MEASURE. 


448. 

■3 


£XA9IPI^S  FOR   PRACTICE. 

Reduce  3  mi.  27  ch.  19  1.  4  in.  to  inches. 


Reduce  14841  1.  to  higher  denominations. 


3.  Reduce  \^  of  a  chain  to  lower  denominations. 

4.  Reduce  .953  of  a  mile  to  links. 

5.  A  lot  having  a  frontage  of  4  rods  contains  ^  of  an  acre.  What  is  its  depth 
in  chains,  links,  and  inches? 

6.  A  field  37  ch.  42  1.  long,  and  30  cli.  21  1.  Avide,  will  require  li6w  many  feet 
of  fence  to  enclose  it? 

7.  How  many  rods  of  fence  wire  will  enclose  a  farm  "il  ch.  50  1.  long  and 
18  ch.  60  1.  wide,  if  tlie  fence  be  made  6  wires  high  ? 

8.  A  garden  is  307f  feet  long  and  250|  feet_wide.  What  is  tlie  girt,  in 
chains,  links,  and  inches,  of  a  wall  surrounding  it  ? 

9.  An  errand  boy  goes  from  his  starting  point  east  33  ch.  50  1.  3  in.,  thence 
north  14  ch.  90  1.  2  in.,  and  returns.  How  many  full  steps  of  2  feet  4  inches 
did  he  take,  and  what  was  the  remaining  distance  in  inches  ? 


SURVEYOR'S   SQUARE    MEASURE. 
449.     The  ITnit  of  land  measure  is  the  acre. 

Table. 

625  square  links  (sq.  1.)  =  1  square  rod sq.  rd. 

16  square  rods =1  square  chain. .  sq.  ch. 

10  square  chains,  or  )    _  ^  Y 

160  square  rods f   "~ 

640  acres =1  square  mile sq.  mi.  ' 

Remark. — In  surveying  United  States  lands,  a  selected  Korth  and  South  line  is  surveyed  as 
a  Principal  Meridian,  and  an  East  and  West  line,  intersecting  this,  is  surveyed  as  a  Base  Line. 
From  these,  other  lines  are  run  at  right  angles,  six  miles  apart,  which  divide  the  territory  into 
Townships  six  miles  square. 

The  surface  of  the  earth  being  convex,  these  merid- 
ians converge  slightly.  The  towu.ships  and  sections 
are,  therefore,  not  perfectly  rectangular;  thus  is  cre- 
ated the  necessity  for  occasional  offsets  called  Cor- 
rection Lines. 

Each  township  (Tp.)  is  divided  into  36  equal 
squares  of  1  square  mile  each,  as  shown  in  the 
first  diagram.  These  squares  are  called  sections 
(Sec),  and  are  divided  into  halves  and  quarters;  each 
quarter-.section,  160  acres,  is  in  turn  divided  into 
halves,  or  lots  of  80  acres,  and  quarter  or  half  lots 
of  40  acres  each,  as  shown  in  the  second  diagram. 

The  row  of  townships  running  north  and  south  is 

called  a  Range;   the  townships   in   each    range   are 

numbered  north  and  south  from  the  base  line,  and 

Township.  the  ranges  numbered  east  and  west  from  the  principal 


CUBIC    MEASURE. 


143 


N. 
1  Mile. 


g^ 


N.  }4  Section. 
320  Acres. 


8.  W.  )i  Sec. 
160  A. 


W.^of 

S.E.J^ 
Sec. 

80  A. 


N.  B.  % 
of  S.  E. 

40  A. 


S.  E.  H 
of  S.  E 

40  A. 


8. 

Section. 


meridian.  The  numbering  of  the  sections  in  every 
township  is  as  in  the  township  diagram  given,  and 
the  corners  of  all  quarter-sections  are  permanently- 
marked  by  monuments  of  stone  or  wood,  and  a 
description  of  each  monument  and  its  location  (sur- 
roundings) made  in  the  field  notes  of  the  surveyor. 

The  advantages  of  the  United  States  survey  over 
all  others  are:  1st,  its  official  character  and  uni- 
formity; and  2d,  its  simplicity.  Any  one  having  a 
sectional  map  of  the  United  States  may  place  a  pencil 
point  upon  any  described  land,  thus  knowing  abso- 
lutely its  exact  location. 

For  example,  Sec.  26,  Tp.  24,  N.  of  Range  8,  E.  of 
the  5th  Principal  Meridian,  describes  a  section  in 
the  24th  tier  of  townships  north  of  the  base  line, 
and  8th  range  east  of  the  fifth  principal  meridian. 


EXAMPLES   FoA  PRACTICE. 

450.     i.     Make  a  diagram  of  a  township,  and  locate  S.  ^  of  Sec.  21,  and 
mark  its  acreage. 

2.  Make  a  diagram  of  a  township,  and  locate  S.  E.  ^  of  See.  16,  and  mark 
its  acreage. 

3.  Make  a  diagram  of  a  township,  and  locate  N.  W.  ^  of  S.  W.  i  of  Sec.  12, 
and  mark  its  acreage. 

4.  Make  a  diagram  of  a  township,  and  locate  Sees.  35,  26,  and  E.  4  of  27, 
and  mark  their  acreage. 


CUBIC   MEASURE. 


-^ 

/ 

/ 

;FOOT 

3  FT. 


N 

V         V       - 

\v^ 

V         \           V         X^ 

\ 

\ 

\      \      \ 

\ 

451.  Cubic  Measure  is  used  in  measuring  solids  or 
volume. 

452.  A  Solid  is  that  which  lias  length,  breadth,  and 
thickness  ;  as  the  walls  of  Ijuildings,  bins  of  grain,  timber, 
wood,  stone,  etc. 


453.  A  Cube  is  a  regular  solid  bounded 
by  six  equal  square  sides,  ot  faces  ;  hence  its 
length,  breadth,  and  thickness  are  equal. 

454.  The  Measuring  Unit  of  solids 

is  a  cube,  the  edge  of  which  is  a  linear  unit. 
Thus  a  cubic  foot  is  a  cube,  each  edge  of 
Avhich  is  1  foot ;  a  cubic  yard  is  a  cube,  eacli 
edge  of  which  is  1  yard.  See  the  accom- 
panying diagrams. 


CUBIC    YARD 


144 


CUBIC    MEASIRE. 


455.     To  Find  the  Volume  of  a  Solid. 

Rule. — Multiply  iogethei    its  IcngtJt,  hicadth,  and  tlii^ikueiis. 

Table. 

1728  cubic  inches  (cu.  in.)  =  1   cubic  foot cu.  ft. 

27  cubic  feet.. =  1  cubic  yard cu.  yd. 

128  cubic  feet .     =  1   cord  of  wood  ..cd. 

Special  Cubic  Measures. 

100  cubic  feet  =  1  register  ton  (shipping). 
40  cubic  feet  =  1  freight  ton  (shipping). 
16^  cubic  feet  =  1  perch  of  masonry. 


456.    A  Cord    of 

■wood  is  a  pile  8  feet 
long,  4  feet  -wide,  and 
4  feet  high. 

457.    A  Cord  Foot 

g  is  one  foot  in  length  of 
sucli  a  pile. 


458.  To  Find  the  Cubical  Contents  of  Square  Timber. 

Bule. — Multiply  together  the  feet  ineasurements  of  length,  width,  and 
depth . 

459.  To  Carry  Timbers,  one  person   supporting  an  end  and  two  others  with 
bar. 

DiKFxnoNS. — Let  the  tiro  with  the  bar  lift  at  a  point  J  the  length  from  the  end. 

REM.VRK. — 1.  Formerly  a  perch  of  masonry  was  24|  cu.  ft.;  but  the  perch  of  16|  cu.  it., 
which  is  16^  ft.  long,  1  ft.  high,  and  1  ft.  wide,  is  now  in  general  use. 

2.  A  cubic  yard  of  earth  is  called  a  load. 

3.  Mechanics  estimate  their  work  on  walls  by  the  girt,  and  no  allowance  is  made  for  windows 
or  doors.     In  estimating  the  amount  of  material  required,  such  allowances  are  made. 

Formulas  for  Rectaugular  Solids. 

Lemjtlt  X  Breadth  x  Higlit         =  Volume. 
Volume  H-  {Length  X  Breadth)  =  Eight. 
Volume  -=-  {Length  X  Sight )      =  Breadth. 
Volume  -=-  (Breadth  x  Hight)    =  Length. 

Remark. — The  three  given  dimensions  must  be  expressed  in  units  of  the  same  denomination. 

46<).    To  Find  the  Number  of  Bricks  for  a  WaU 
^\\U.— Multiply  the  cubic  feet  by  23%,  and  add  %U. 

Rem.\rk. —  For  guide  in  purchasing  material  the  above  will  be  found  correct  for  bricks 
8  in.  X  4  in.  x  2  in.,  after  allowing  for  mortar. 


EXAMPLES    FOR    PRACTICE.  145 

461.    To  Find  the  Number  of  Perches  in  a  Wall. 

Rule. — Divide  the  contents  of  the  wall,  in  feet,  hy  16H. 

EXAIMTPLES   FOK  PKACTICE. 

462.     -?.     Reduce  468093  cu.  in.  to  higher  denominations. 
3.     Reduce  132  cu.  yd.  11  cu.  ft.  981  cu.  in.  to  cubic  inches. 

3.  What  is  the  volume  of  a  solid  8  ft.  3  in.  long,  5  ft.  10  in.  high,  and  4  ft. 
•6  in.  wide  ? 

4.  How  many  cubic  feet  of  air  in  a  room  26  ft.  8  in.  long,  22  ft.  6  in.  wide, 
and  12  ft.  high  ? 

5.  How  many  cubic  yards  of  earth  must  be  removed  in  digging  a  cellar  60  ft. 
long,  30^  ft.  wide,  and  7^  ft.  deep  ? 

6.  How  many  perches  of  masonry,  of  16^  feet  each,  in  a  wall  85  ft.  long,  32 
ft.  high,  and  li  ft.  thick  ? 

7.  Reduce  -|  of  a  cubic  inch  to  the  fraction  of  a  cubic  yard. 

8.  What  decimal  part  of  a  cubic  yard  is  7  cu.  ft.  108  cu.  in.  ? 

9.  What  fraction  of  a  cubic  foot  is  220  cu.  in.  ? 

10.  Reduce  .525  of  a  cubic  yard  to  lower  denominations. 

11.  What  will  be  the  cost,  at  21(^'  i)er  cubic  yard,  of  excavating  for  a  reser- 
voir 180  ft.  long,  105  ft.  3  in.  wide,  and  15  ft.  9  in.  deep  ? 

12.  What  will  be  the  cost  of  building  the  walls  of  a  block  140  ft.  long,  66  ft. 
wide,  and  57  ft.  high,  at  $1.40  per  perch  of  16^  cu.  ft.,  if  the  wall  is  16  in. 
thick,  and  no  allowance  bo  made  for  openings  ? 

13.  How  many  common  bricks  will  be  required  for  the  above  wall,  allowance 
being  made  for  28  windows  each  3^  ft.  wide  and  8  ft.  high,  48  windows  each 
3  ft.  9  in.  wide  and  8  ft.  high,  and  4  doors  each  8  ft.  wide  and  11  ft.  high  ? 

14-.  A  room  28  ft.  long,  18  ft.  wide,  and  12  ft.  high,  will  store  how  many 
•cords  of  wood  ? 

lo.  How  many  cords  of  wood  in  a  pile  108  ft.  long,  7  ft.  9  in.  high,  and 
•6  ft.  wide  ? 

16.  From  a  i)ilo  of  wood  71  ft.  6  in.  long,  9  ft.  4  in.  wide,  and  6  ft.  8  in. 
bigh,  21f  cords  were  sold.     What  was  the  length  of  the  pile  remaining  ? 

17.  At  $4.75  per  cord,  what  will  it  cost  to  fill  with  wood  a  shed  34  ft.  long, 
18  ft.  wide,  and  10  ft.  high  ? 

18.  What  is  the  weight  of  a  block  of  granite  11  ft.  3  in.  long,  3  ft.  G  in. 
thick,  and  8  ft.  4  in.  wide,  if  it  weiglis  166  lb.  per  cubic  foot  ? 

19.  What  is  the  weight  of  a  white  oak  timber  15  in.  square  and  40  ft.  long, 
if  the  weight  per  cubic  foot  be  72.5  lb.  ? 

20.  How  many  cubes  1  in.  on  each  edge  can  be  cut  from  a  cubic  yard  of 
wood,  if  no  allowance  be  made  for  waste  by  sawing  ? 

21.  Find  the  contents  of  a  cube,^  each  edge  of  which  is  2  yd.  7^  in. 

22.  How  many  perches  of  masonry  in  a  wall  7^  ft.  high  and  2  ft.  thick, 
•enclosing  a  yard  12J  rods  long  and  9^  rods  wide  ?  How  many  bricks  will  be 
required,  and  if  bricks  cost  $6.50  per  M  and  laying  them  cost  $1.60  per  M, 
"what  will  be  the  cost  of  the  wall  ? 

10 


i46  producers'  and  dealers'  approximate  rules. 

23.     What  is  the  volume  of  a  rectangular  solid  11  ft.  long,  -i^  ft.  wide,  and 
4  ft.  high  ? 

2j^     a  cask  holding  %bQ\  gal.  of  water  will  hold  how  many  bushels  of  wheat?" 


PRODUCERS'  AND  DEALERS'  APPROXIMATE  RULES. 

163,  To  find  the  contents  of  a  bin  or  elevator  in  bushels,  stricken  measure. 
Rule. — Miiltiply  the  cubic  feet  bj/.S,  and  add  1  bushel  for  each  300,  or  in  that 
proportion. 

To  find  the  contents  of  a  bin  or  crib  in  bushels,  by  heaped  measure. 
Rule. — Multiply  the  cubic  feet  by  .63. 

Remark. — If  the  crib  jlare,  take  the  mean  width. 

To  find  the  number  of  shelled  bushels  in  a  space  occupied  by  unshelled  com^ 
Rule. — Divide  the  cubic  inches  by  SSJfO,  or  multiply  the  cubic  feet  by  Jf5. 

To  find  the  dimensions  of  a  bin  to  hold  a  certain  number  of  bushels. 
Rule. — To  the  number  of  bushels  add  one-fourth  of  itself,  and  the  sutn  will  be 
the  cubic  feet  required,  to  loithin  one  three-hundredth  part. 

To  find  the  exact  number  of  stricken  bushels  in  a  bin.  Rule. — Divide  the 
cubic  inches  by  2150.42. 

To  find  the  exact  number  of  heaped  bushels  in  a  bin.  Rule. — Divide  the 
cubic  inches  by  2747. 71. 

To  find  the  capacity  of  circular  tanks,  cisterns,  etc.  Rule. — The  square  of 
the  diameter,  multiplied  by  the  depth  in  feet,  ivill  give  the  number  of  cylindrical 
feet.     Multiply  by  5^  for  gallons,  or  multiply  by  .1865  for  barrels. 

Remark. — In  tanks  or  casks  having  bilge,  find  the  mean  diameter  by  taking  one-half  of  the 
Bum  of  the  diameters  at  the  head  and  bilge. 

To  find  the  number  of  perches  of  masonry  in  a  wall,  of  24f  cubic  feet  in  a 
perch.     Rule. — Multiply  the  cubic  feet  by  .0404. 

To  find  the  number  of  perches  of  masonry  in  a  wall,  of  1(J^  cubic  feet  in  a 
perch.     Rule. — Multiply  the  cubic  feet  by  .0606. 

Remark.— The  above  is  correct  within  ^^^^  part.     In  large  contracts  add  -^^  of  1%. 

Example.  —  How  many  perches,  of  24f  cu.  ft.  each,  in  a  wall  150  ft.  long,  50 
ft.  high,  and  3  ft.  thick? 

Explanation.— 5A<^<  Metliod.—lbQ  X  50  x  2  =  15000;  15000  x  .0404  =  606;  add  y^,  or 
.606  =  606.606. 

Extended  Method.— 150  x  50  X  2  =  15000;  15000  h-  24.75  =  606.6,  same  as  before. 

Same  example,  perch  of  10^  cu.  ft. 

Explanation.  --SAorf  Met/u>d.— 150  x  50  x  2  =  15000;  15000  x  .0606  =  909;  add  j^'^^  =  .9; 
909  +  .9  =  909.9. 

To  find  the  number  of  cubic  feet  in  a  log.  Rule. — Divide  the  average 
diameter  in  inches  by  3,  square  the  quotient,  multiply  by  the  length  of  the  log 
in  feet,  and  divide  by  36. 


CUBE    ROOT.  147 

To  find  the  number  of  feet,  board  measure,  in  a  log.  Rule. — Multiply  the 
cubic  feet,  as  above  obtained,  by  9. 

HAY   MEASUREMENTS. 

464.  Few  products  are  so  difficult  of  accurate  measurement  as  hay,  owing  to 
the  pressure,  or  the  want  of  it,  in  packing,  time  of  settling,  volume  in  bulk,  and 
freedom  from  obstruction  in  packing.  Plainly,  the  larger  (higher)  the  stack,  or 
mow,  and  the  greater  the  foreign  weight  in  compress,  the  more  comi)act  it  will  be. 

465.  The  accepted  measurements  are  of  three  kinds: 

1st.  To  find  the  weight  of  hay  in  a  load  or  shed  loft,  unpressed.  Eule. — Allow 
5Jfi  cubic  feet  for  a  ton. 

2d.  TofindthcAveight  in  common  hay  barn,  or  small  (low)  stack.  Eule.— ^l/Zo?/' 
Jfi5  cubic  feet  for  a  ton. 

3d.  To  find  the  Aveight  in  mow  bases  in  barns,  compressed  with  gram,  and 
in  butts  of  large  stacks  of  timothy  hay.     Eule. — Alloir  32Jf.  cubic  feet  for  a  ton. 

CUBE   ROOT. 

466.  The  Cube  or  Third  Power  of  a  number,  is  the  product  of  three  equal 
factors. 

467.  The  Cube  Root  of  a  number  is  one  of  the  three  equal  factors  the 
product  of  which  represents  the  cube.  Thus,  a  cubic  foot  =  13  X  12  X  12,  or 
1728  cubic  inches,  the  product  of  its  length,  breadtli,  and  thickness;  and  since 
12  is  one  of  the  three  e(|ual  factors  of  1728,  it  must  be  its  cube  root. 

468.  The  operation  of  finding  one  of  the  equal  factors  of  a  cube  is  called 
extracting  the  ciibe  root. 

469.  As  shown  in  the  explanation  of  extracting  the  square  root,  the  first 
point  to  be  settled  in  extracting  any  root  is  the  relative  number  of  unit  orders 

^or  places  in  the  number  and  its  root. 

470.  XJnit^  and  Cubes  Compared. 


Remauk. — From  this  comparison  may  be  inferred  the  following: 

1st.  The  cube  of  any  number  expressed  by  a  single  figure  cannot  have 
less  than  one  nor  more  than  three  places  or  unit  ordeis. 

2d.  Each  place  added  to  a  number  will  add  three  places  to  its  cube. 

3d.  If  a  number  be  separated  into  periods  of  three  figures  each,  begin- 
ning at  the  right  hand,  the  number  of  places  in  the  root  will  equal  the 
number  of  periods  and  partial  periods  if  there  are  any. 

10'  =  1000 

471.     To  help  in  understanding  the  cube  root,  first  form  a  cube  and  thus 
ascertain  its  component  parts  or  elements.     Take  57  as  the  number  to  ])e  cubed. 


U.MTS. 

Cubes. 

1' 

= 

1 

2* 

= 

8 

3* 

= 

27 

■i' 

= 

64 

5' 

= 

125 

C 

= 

21G 

=: 

343 

8* 

= 

512 

9' 

= 

729 

148 


CUBE   ROOT. 


(50*  X 
+  2  (50' 

50» 

50  +  7 
50  +  7 
(50  X  7)  +  7' 
+  (50  X  7) 

50^4- 

-  2  (50  X  7)  +  7'^ 
50  +  7 

50» 

7)  + 
X7) 

2  (50  X  7")  +  r 
+  (50  X  T) 

Explanation. —Cubing  57,  we 
have  57  x  57  x  57  =  185193;  or, 
separating  57  into  its  tens  and  units 
gives  5  tens  or  50 -f  7  units;  or, 
50  -\-  7.  Cube  the  given  number, 
by  using  it  in  this  form  three  times 
as  a  factor,  and  the  result  is  185193. 


50'  +  3  (50*  X  7)  +  3  (50  X  7')  +  7=  =  185193 

472.  From  this  result  observe  that  57^  =  the  cube  of  the  tens,  plus  three 
times  the  square  of  the  tens  multiplied  by  the  units,  plus  three  times  the  tens, 
multiplied  by  the  square  of  the  units,  plus  the  cube  of  the  units;  or  that  the 
cube  of  any  number  made  up  of  tens  and  iinifs  =  t'  +  3t'  u  +  3  t  u'  +  \i\ 
which  for  the  purpose  of  reference  we  will  call  Formula  {a).  And  if  all  orders 
above  simple  units  are  considered  tens,  Formula  {a)  will  apply  to  the  cube  of  any 
number. 

•473.  To  assist  in  understanding  the  operation  of  extracting  the  cube  root, 
observe  the  forms  and  dimensions  of  the  illustrative  blocks,  and  the  relation  of 
each  to  the  other  in  the  formation  of  the  complete  cube. 


•    Operation 

t    u 
t3  +  3t2u  +  3tu2  +  u3=185.193  [5    7 

t3=  125 --.or  125000 
StMi +  3tu2+ u3=     60193  =  rem. 


3t: 


t  =  50 
t2  =  25U0 
3  t2  =  7500 
3t  =    150 
+  3  t  =  7650  trial  divisor. 


3t2u 
3tu2 


u^  = 


52500 

7350 

343 


3  t2  u  +  3  t  u2  +  u3  =  60193. 


Explanation. — Since  the  block  (A) 
is  a  cube,  the  number  representing  the 
length  of  its  side  will  be  its  cube  root. 

The  given  number  consists  of  two 
periods  of  three  figures  each,  therefore 
its  cube  root  will  contain  two  places, 
tens  and  units. 

Since  the  given  number  is  a  product 
of  its  root  taken  three  times  as  a  factor, 
the  first  figure,  or  highest  order  of  the 
root,  must  be  obtained  from  the  first 
left  hand  period,  or  highest  order  of  the 
power;  therefore,  find  first  the  greatest 
cube  in  185;  since  185  comes  between 
125  (the  cube  of  5)  and  216  (the  cube  of 
6)  the  tens  of  the  root  must  be  5  plus  a 
certain  remainder;  therefore,  write  5  in 
the  root  as  its  tens  figure. 

Subtracting  the  cube  of  the  root 
figure  thus  found  (5  tens,  or  50)'  = 
125000,  by  taking  125  from  the  left 
hand  period,  185,  and  so  obviate  the 
necessity  of  writing  the  ciphers ;  to 
this  remainder  bring  down  the  next,  or 
right  hand  period,  193,  thus  obtaining 
as  the  entire  remainder  60193. 

Referring  to  Formvla  (a),  observe 
that,  having  subtracted  from  the  given 
number  the  cube  of  its  tens  ( t' ),  the 
remainder,  60193,  must  be  equal  to  3  t' 
u  +  3tu*  +  u''. 


CUBE    ROOT. 


U9 


If  a  cube  (B),  50  inches  in  length  on  each  side,  h  fonned,  its  contents  will  equal  125000 

cubic  inches,  and  it  will  be  shown  that  the 
remaining  60193  cubic  inches  are  to  be  so 
added   to   cube  (B)    that   it   will   retain   its 
cubical  form.     In  order  to  do  this,  equal  ad- 
ditions must  be  made  to  three  adjacent  sides, 
and  these  three  siiles,  being  each  50  inches  in 
length  and  50  inches  in  width,  the  addition  to 
each  of  them  in  surface,  or  area,  is  50  ^  50, 
or  50-,  and  on  the  three  sides,  3  (50*),  or  3  t«, 
as  in  the  squares  (C).    It  will  also  be  observed 
that  three  oblong  blocks  (D)  will  be  required 
to  fill  out  the  vacancies  in  the  edges,  and  also 
the  small  cube  (E),  to  fill  out  the  corner. 
Since  each  of  the  oblong  blocks  has  a  length  of 
5  tens,  or  50,  inches,  the  three  will  have  a  length  of 
3  X  50  inches,  or  3  t.     Observe,  now,  the  surface  to 
be  added  to  cube  (B),  in  order  to  include  in  its  con- 
tents the  60193  remaining  cubic   inches,  has  been 
nearly,  but  not  exactly  obtained  ;   and  since  cubic 
contents  divided  by  surface  measurements  must  give 
units  of  length,  the  thickness  of  the  three  scjuares 
(C),  and  of  the  three  oblong  pieces  (D),  will  be  de- 
termined by  dividing  60193  by  the  surface  of  the 
three  squares,  plus  the  surface  of  tbe  three  oblong 
blocks,  or  by  3  t-  +  3  t ;   this  division  may  give  a 
quotient  too  large  owing  to  the  omission  in  the  di^^• 
sor  of  the  small  square  in  the  corner;   hence  such 
surface  measure  taken  as  a  divisor,  may  with  pro- 
priety, be  called  a  trial  divisor.     So  using  it,  7  is 
obtained  as  the  second,  or  unit  figure  of  the  root. 

Assuming  this  7  to  be  the  thickness  of  the  three 
square  blocks  (C),  and  both  the  hight  and  thickness 
of  the  three  oblong  blocks  (D),  gives  for  the  solid 
contents  of  the  three  square  blocks  (C),  52500,  and 
for  the  solid  contents  of  the  three  oblong  blocks  (D), 
7350,  or  3  t-  u-f  3  t  u-  =  59850 ;  and  by  reference 
to  the  Formula  (a),  observe  that  the  only  term  or  ele- 
ment required  to  complete  the  cube  of  (t  -f  u)  is  the 
cube  of  the  units  (u  ^). 

Now,  by  reference  to  the  illustrative  blocks,  observe 
that  by  placing  the  small  cube  (E)  in  its  place  in  the 
corner,  the  cube  is  complete.  And  since  (E)  has  been 
found  to  contain  7  X  7  X  7,  or  343  cubic  inches,  add 
this  to  the  sum  of  3  t-  u  3  tu-  u^  and  obtain  3  t-  u 
-f  3  t  u2  +  u»  =  60193;  and  if  to  this  t\  or  125000 
is  added,  the  result  is  t»  +3  t  -u-f-3  t  u«  -|-u»  = 
185193,  Forimdn  (a);  then  subtracting  60193  from 
the  remainder,  60193,  nothing  remains. 

This  proves  that  the  cube  root  of  185193  is  57.  By 
the  operation  is  also  proved  the  correctness  of  Form- 
ula {a)  :  The  cube  of  any  number  equals  the  cube  of 
its  tens,  plus  three  times  the  square  of  its  tens  mul 
tiplied  by  its  units,  plus  three  times  its  tens  multiplied  by  the  square  of  its  units,  plus  the  cube 
of  its  units. 


150  EXAMPLES   FOR    PRACTICE. 

Bule. — I.  Beginning  at  the  right,  separate  the  given  number  into 
periods  of  three  figicres  each. 

II.  Take  for  the  first  root  figure  the  cube  root  of  the  greatest  perfect 
cube  in  the  left  hand  period;  subtract  its  cube  from  this  left  hand 
period,  and  to  the  remainder  bring  down  the  next  period. 

in.  Divide  this  remainder,  using  as  a  trial  divisor  three  times  the 
square  of  the  root  figure  already  found,  so  obtaining  the  second  or  units 
figure  of  the  root;  next,  subtract  from  the  remainder  three  times  the 
square  of  the  tens  muUiplied  by  the  units,  plus  three  times  the  tens 
multiplied  by  the  square  of  the  units,  plus  the  cube  of  the  units. 

Remarks. — 1.  In  examples  of  more  periods  than  two,  proceed  as  above,  and  after  two  root 
figures  are  found,  treat  both  as  tens  for  finding  the  third  root  figure.  For  finding  subsequent 
root  figures,  treat  all  those  found  as  so  many  tens. 

2.  In  case  the  remainder,  at  any  time  after  bringing  down  the  next  period,  be  less  than  the 
trial  dirUor,  place  a  cipher  in  the  root  and  proceed  as  before. 

3.  Should  the  cube  root  of  a  mixed  decimal  be  required,  form  periods  from  the  decimal  point 
right  and  left.  If  the  decimal  be  pure,  point  off  from  the  decimal  point  to  the  right,  and  if 
need  be  annex  decimal  ciphers  to  make  periods  full. 

4  To  obtain  approximate  roots  of  imperfect  cubes,  to  any  desired  degree  of  exactness, 
annex  and  use  decimal  periods. 

5.  The  cube  root  of  a  common  fraction  is  the  cube  root  of  its  numerator  divided  by  the  cube 
root  of  its  denominator. 

6.  The  cube  root  of  any  common  fraction  may  be  found  to  any  desired  degree  of  exactness, 
either  by  extracting  the  root  of  its  terms  separately  (adding  decimal  periods  if  need  be)  or  by 
first  reducing  the  common  fraction  to  a  decimal  and  then  extracting  the  root. 

7.  The  4th  root  can  be  obtained  by  extracting  the  square  root  of  the  square  root. 

8.  The  6th  root  is  obtained  by  taking  the  cube  root  of  the  square  root,  or  the  square  root  of 
the  cube  root. 

EXAMPLES   FOB  PKACTICE. 

4:74-.     1.     What  is  the  cube  root  of  1728  ? 

2.  What  is  the  cube  root  of  15625  ? 

3.  What  is  the  cube  root  of  110592  ? 
Jf.     What  is  the  cube  root  of  65939204  ? 

5.  Find  the  cube  root  of  2146,  to  three  decimal  places. 

6.  Find  the  cube  root  of  119204,  to  two  decimal  places. 

7.  Find  the  cube  root  of  46982,  to  one  decimal  jilace. 

8.  Find  the  cube  root  of  ^^, 

9.  Find  the  cube  root  of  y^-^. 

10.  Find  the  cube  root  of  -g-^^mH^. 

11.  Find  the  cube  root  of  ^\\,  to  one  decimal  place. 

12.  Find  the  cube  root  of  \l^^,  to  two  decimal  places. 

15.  Find  the  cube  root  of  25.41G23T,  to  two  decimal  places. 
IJf.     Find  the  cube  root  of  3496.25,  to  three  decimal  places. 

16.  Find  the  cube  root  of  .4106,  to  three  decimal  places. 

16.  Find  the  decimal  equivalent  of  the  cube  root  of  \\,  to  two  decimal  i)lace6, 
by  reducing  tlie  fraction  to  a  decimal  of  six  places  and  extracting  the  root  of  the 
decimal. 


MISCELLANEOUS  MEASUREMENTS.  151 

ir.     What  must  be  the  liiglit  of  a  cubical  bin  that  will  hold  1000  bu.  of  wheat? 

18.  The  width  and  hight  of  a  crib  of  unshelled  corn  are  equal,  and  each  is 
one-third  of  its  length.  If  the  contents  of  the  crib  are  7465  bushels,  what  is 
its  length  ? 

19.  If  the  hight  of  an  oat  bin  is  twice  its  width,  and  its  length  is  three  and  one- 
half  times  its  hight,  what  must  be  its  dimensions,  if  the  bin  holds  1750  bushels? 

20.  A  cubical  cistern  contains  630  barrels.     How  deep  is  it  ? 

21.  A  square  cistern,  the  capacity  of  which  is  420  barrels,  has  a  depth  equal 
to  onlv  one-hftlf  its  width.     Find  its  dimensions. 


DUODECIMALS. 

475.  Duodecimals  are  denominate  fractious  of  either  linear,  square,  or 
■cubic  measure.  They  are  found  by  successive  divisions  of  the  unit  by  12,  and 
are  added,  subtracted,  multiplied,  and  divided  in  the  same  manner  as  compound 
numbers,  though  they  may  be  treated  as  fractions,  12  being  the  uniform  denom- 
inator.    The  scale  is  uniformly  12. 

476.  The  Unit  of  measure  in  Duodecimals  is  the  foot.  Its  first  division  by 
12  g'wes prinies  ( ' );  primes  divided  by  12  give  seconds  (  "  ),  seconds  divided  by 
12  give  thirds  (  '"  ),  and  so  on. 

Remark. — Duodecimals  are  but  little  used. 


MISCELLANEOUS    MEASUREMENTS. 
477.     A  Triangle  is  a  plane  figure  bounded  by  three  straight  lines. 

478  To  find  the  area  of  a  triangle,  the  base  and  hight  being  given. 
KuLE. — MuUiph/  the  base  by  one-half  the  hight. 

To  find  the  area  of  a  triangle,  when  the  three  sides  are  given.  Rule. — Find 
one-half  of  the  sum  of  the  three  sides;  from  this  subtract  each  side  separately; 
multi2)ly  together  the  four  results  thus  obtained,  and  extract  the  square  root  of  the 
product. 

To  find  the  area  of  any  plane  figure,  the  ojiposite  sides  of  which  ara  equal  and 
parallel.     Rule. — Multiply  the  base  by  the  perpendicidar  hight. 

To  find  the  area  of  a  plane  figure,  Avhose  opposite  sides  are  i)arallel  but  of 
unequal  length.  Rule. — Obtain  the  average  length,  and  multiply  by  the  per- 
piendicular  hight. 

479.  A  Circle  is  a  plane  figure  bounded  by  a  curved  line, 
every  part  of  which  is  equally  distant  from  a  i)oint  within 
called  the  center. 

480.  The  Circumference  of  a  circle  is  the  curved  line 
boundinsj  it. 


152 


MISCELLANEOUS   MEASUREMENTS. 


481.  The  Diameter  of  a  circle  is  a  straiglit  line  i)assiug  through  tlie  center 
and  terminating  in  the  circumference. 

482.  The  Radius  of  a  circle  is  a  straight  line  i)as8ing  from  the  center  to 
any  point  of  the  circumference. 

483.  To  find  the  circumference  of  a  circle,  the  diameter  being  given. 
Rule. — Multiply  the  diameter  hy  S.lJflG. 

To  find  the  diameter  of  a  circle,  the  circumference  being  given.  Rule. — Divide 
the  circumference  hy  S.lJflG. 

To  find  the  area  of  a  circle,  the  circumference  and  diameter  being  given. 
Rule. — Multiply  the  circumference  hy  the  diameter,  and  divide  the  product  hy  4- 

To  find  the  side  of  a  square  equal  in  area  to  a  given  circle.  Rule. — Multiply 
the  circumference  hy  .2821. 

To  find  the  area  of  a  square  that  can  be  inscribed  within  a  given  circle. 
Rule. — Mulfijily  the  square  of  the  radius  hy  2,  and  extract  the  square  root  of  the 
result. 

484.  A  Cylinder  is  a  circular  body  of 
uniform  diameter,  the  ends  of  which  are 
parallel  circles. 

Remahk. — The  convex  surface  of  a  cylinder  is 

equal  to  the  surface  of  a  rectangular  body,  the  length 

and  hight  of  which  are  equal  to  the  circumference 

and  hight  of  the  cylinder.     See  the  figure,  A,  B,  C, 

^,  D,  back  of  the  cylinder  in  the  acompanying  diagram. 

CYLINDER  AND  RECTANGLE. 

485.  To  find  the  surface  or  area  of  a  cylinder.  Rule. — Multiply  the  cir- 
cumference hy  the  hight. 

.  To  find  the  contents  of  a  cylinder.     Rule. — Multiply  the  area  of  the  base  by 
the  hight. 


486.  A  Pyramid  is  a  solid,  the 
base  of  Avhich  has  three  or  more  equal 
sides,  terminating  in  a  point  called  a 
vertex. 

487.  A  Cone  is  a  solid  which  has 
a  circular  base,  its  convex  surface  ter- 
minating in  a  point  called  a  vertex. 


PYRAMID. 


488.  To  find  the  surface  of  a  regular  pyramid  or  cone.  Rule. — Multiply 
the  perimeter  or  circumference  of  the  base,  by  one-half  the  slant  hight. 

To  find  the  contents  of  a  i)yramid  or  cone.  Rule. — Multiply  the  area  of  the 
base  hy  one-third  the  perpendicular  hight. 


EXAMPLES    FOR    PRACTICE.  153 

489.  A  Sphere  is  a  solid  bounded   by  a  curved 
surface,  all  points  of  Avhich  are  equally  distant  from  a 

A        point  within  called  the  center. 

490.  The  Diameter  of  a  sphere  is  a  line   drawn 
through  its  center,  terminating  each  way  at  the  surface. 

491.  To  find  the  surface  of  a  sphere.  Rule. — Multiply  the  square  of  its 
diameter  by  3.  H16. 

To  find  the  volume  of  a  sphere.  Rule. — Multiply  the  cuhe  of  the  diameter 
hy  .5236. 

To  find  how  large  a  cube  may  be  cut  from  any  given  sphere,  or  may  be 
inscribed  within  it.  Rule. — Divide  the  square  of  the  diatneter  of  the  sphere  by 
3,  and  extract  the  square  root  of  the  quotient;  the  root  thus  found  will  be  the 
length  of  one  side  of  the  cube. 

To  gauge  or  measure  the  capacity  of  a  cask.  Rule. — Multiply  the  square  of 
the  mean  diameter  in  inches  by  the  length  in  inches,  and  this  product  by  .003 Jf.; 
the  result  will  be  the  capacity  in  gallons. 

Remark. — In  case  the  cask  is  only  partly  full,  stand  it  on  end,  find  the  mean  diameter  of 
the  part  filled,  multiply  its  square  by  the  hight,  and  that  product  by  .0034. 

EXAMPLES  FOR    PRACTICE. 

Remark. — In  giving  one  example  under  each  of  the  several  preceding  rules  in  measure- 
ments, the  object  is  as  much  for  reference  as  for  practice  in  solving. 

492  1.  How  many  square  feet  in  the  gable  end  of  a  house  24  ft.  wide  and 
6  ft   6  in.  high  ? 

2.  Find  the  number  of  square  yards  in  a  triangular  sail,  the  sides  of  Avhich 
arc  36  ft.,  45  ft.,  and  48  ft.  respectively. 

3.  How  many  acres  in  a  rectangular  field  108  rods  long  and  48  rods  wide  ? 

Jf.     A  farm  stretches  across  an  entire  section,  being  200  rods  wide  on  the  west 
line  and  160  rods  wide  on  on  the  east  line.     How  many  acres  in  the  farm  ? 
5      How  many  feet  of  fence  will  inclose  a  circular  pond  82.5  ft.  in  diameter  ? 

6.  What  is  the  diameter  of  a  circle,  the  circumference  of  which  is  90  rods  ? 

7.  The  diameter  of  a  circular  park  is  50  rods.  How  many  acres  does  the 
park  cover  ? 

8.  What  is  the  side  of  a  square  having  an  area  equal  to  that  of  a  circle  100  ft. 
in  diameter  ? 

9.  What  is  the  largest  square  timber  that  can  be  hewn  from  a  log  42  inches 
in  diameter  ? 

10.  What  will  be  the  cost  of  a  sheet-iron  smoke-stack  40  ft.  high  and  2  ft.  in 
diameter,  at  15^*  per  square  foot  ? 

11.  Find  the  capacity  in  gallons  of  a  tank  14  ft.  deep  and  18  ft.  in  diameter? 

12.  A  pyramid  has  a  triangular  base  3  ft.  on  each  side,  and  a  slant  hight  of 
of  10  ft.     Find  the  number  of  square  feet  in  its  surface. 


154  TABLES    AND    CUSTOMS   IS"   THE    PAPER    AXD    BOOK   TRADE. 

13.     A  tent  is  in  the  form  of  a  cone;  if  its  slant  hight  is  16  ft.  and  its  base 

circumference  30  ft.,  how  many  square  yards  of  duck  were  used  Ju  making  it  ? 

H.     How  many  square  inches  of  leather  will  cover  a  foot  ball  8  in.  in  diameter? 

15.  How  many  cubic  feet  in  the  contents  of  a  globe  4  ft.  in  diameter  ? 

16.  The  diameter  of  the  earth  is  7901  miles,  and  that  of  the  planet  Jupiter 
85390  miles.     How  many  spheres  like  the  earth  are  equal  to  Jupiter  ? 

17.  What  will  be  the  length  of   the  largest  cube  that  can    be  cut  from  a 
sphere  T901  miles  in  diameter  ? 

18.  A  cask  28  in.  at  each  end,  and  34  in.  at  the  bilge,  is  3  ft.  long.     How 
many  gallons  of  water  will  it  hold  ? 

lU.     If  a  cask  24  inches  at  the  chime,  30  inches  at  the  bung  and  3  feet  long, 
is  f  full,  how  many  more  gallons  may  be  put  into  it  ? 


TABLES   AND    CUSTOMS    IN   THE    PAPER   AND    BOOK   TRADE. 
498.     Pajier  in  the  stationery  trade  is  sold  by  the  following 

Table. 

24  sheets =1  quire. 

20  quires =  1  ream. 

2  reams =  1  bundle. 

5  bundles =  1  bale. 

A  bale  contains  200  quires,  or  4800  sheets. 

Remarks. — 1.  In  copying,  &  folio  is  usually  100  words. 

2.   In  type-setting,  an  em  is  the  square  of  the  body  of  a  type,  used  as  a  unit  by  which  to 
measure  the  amount  of  printed  matter  on  a  page. 

49-4.     Books  are  sometimes  classified  by  their  size,  or  the  number  of  pages  in 

a  sheet. 

Xame.  Sheet  folded  into.  Pa^es. 

Folio, 2  leaves, 4 

Quarto,  4to 4  leaves, 8 

Octavo,  8vo. 8  leaves, 10 

Duodecimo,  12mo 12  leaves, 24 

16mo 16  leaves, 32 

18mo 18  leaves, 3G 

24mo 24  leaves, 48 

32mo. 32  leaves, 64 

Table  for  C'ouuting:. 

12  units  =  1  dozen,     i     12  dozen  =  1  gross. 

20  units  =  1  score.       |     12  gross    =  1  great  gross. 

Table  for  Land  and  Lot  Measures. 

104^    feet  square  =  ^^  of  an  acre.  10  rods  X  16  rods  =  1  acre. 


14Tyy  feet  square  =  -^  of  an  acre. 
'2Q^>-^  feet  square  =  1  acre. 


8  rods  X  20  rods  =  1  acre. 
40  yards  X  121  yards  =  1  acre. 


THE   METRIC    SYSTEM. 


155 


THE    METRIC    SYSTEM. 

495.  Tlic  Metric  System  is  a  decimal  system  of  denominate  numbers. 
It  is  in  use  in  nearly  all  the  European  States,  in  South  America,  Mexico,  and 
Egypt.  It  is  also  used  somewhat  in  Asia,  and  is  authorized  by  law  in  the  United 
States;  but  its  use  here  is  so  limited  as  to  justify  only  a  reference  to  it,  and  the 
presentation  of  its  unit  equivalents  in  our  weights  and  measures,  as  a  reference 
for  interested  parties. 

496.  The  Unit  of  Length  and  basis  of  the  system  is  the  Meters  39.37-1- 
inches,  being  one  ten-milliontli  of  the  distance  from  the  equator  to  the  pole. 
The  unit  of  area  is  the  Ar  (A.);  the  unit  of  solidity  is  the  Ster  (S.);  the  unit  of 
weight  is  the  Gram  (G. );  the  unit  of  capacity  is  the  Liter  (L.).  Higher  denom- 
inations are  called  Dek'a  (10),  Hek'to  (100),  Kil'o  (1000),  and  Myr'ia  (10000). 
Lower  orders  ai-e  called  Dec'i  (tenths),  Cen'ti  (hundredths),  Mil'li  (thousandths)- 


Metric  Linear  Table. 

1  cen'ti-me'ter cm    =  yJ-Q  M. 

=  1  dec'i-meter dm   =  ^^-   M. 

=  1  Meter M. 

=  1  dek'a- me'ter Dm  =  10   M. 

=  1  hek'to-me'ter 11  m  =  100M. 

=  1  kil'o-me'ter Km  —  1000  M. 

=  1  myr'ia-me'ter . .  .•. Mm  =  10000  M. 

Remarks. — 1.  All  tables  are  formed  in  a  similar  manner. 

2.  In  naming:  units,  abbreviations  are  commonly  used. 

3.  The  system  being  on  a  decimal  scale,  the  full  mastery  of  the  names  of  the  higher  and 
lower  denominations,  with  unit  equivalents,  will  be  sufficient  for  practical  use. 

497.     An  Act  of   Congress  requires  all  reductions  from  the  Metric  to   the 
common  system,  or  the  reverse,  to  be  made  according  to  the  following 


10  mil'li-me'ters  {mm) 

10  cen'ti-me'ters 

10  dec'i-me'ters 

10  me'ters 

10  dek'a-me'ters 

10  hek'to-me'ters 

10  kil'o-me'ters 


1  inch  =  2.54  centimeters. 
1  foot  =  .3048  of  a  meter. 
1  yard  =  .9144  of  a  meter. 
1  rod  =  5.029  meters. 
1  mile  =  1.6093  kilometers. 


Tables  of  Equivalent.s. 

Linear  Measure. 

1  centimeter  =  .3937  of  an  incli. 
1  decimeter  =  .328  of  a  foot. 
1  meter  =  1.0936  yards. 
1  dekameter  =  1.9884  rods. 
1  kilometer  =  .62137  of  ii  mile. 

Square  Measure. 


i  s(i.  inch  =  G.452  sq.  centimeters. 
1  sq.  foot  =  .0929  of  a  sq.  meter. 
1  sq.  yard  =  .8361  of  a  sq.  meter. 
1  sq.  rod  =  25.293  of  a  sq.  meter. 
1  acre  =  40.47  ars. 
1  sq.  mile  =  259  hektars. 


1  sq.  centimeter  =  .155  of  a  sq.  inch. 

1  sq.  decimeter  =  .1076  of  a  sq.  foot. 

1  sq.  meter  =  1.196  sq.  yards. 

1  ar  =  3. 954  sq.  rods. 

1  hektar  =  2.471  acres. 

1  sq.  kilometer  =  .3861  of  a  S(i.  mile. 


156 


MOXEY    OF   THE   GERMAN   EMPIRE. 


Ctbic  Measure. 


1  cu.  inch  =  1G.38T  cu,  centimeter. 
1  cu.  foot  =  28.317  cu.  decimeter. 
1  cu.  yard  =  .  7645  of  a  cu.  meter. 
1  cord  =  3.624:  ster. 


1  cu.  centimeter  =  .061  of  a  cu.  inch. 
1  cu.  decimeter  =  .0353  of  acu.  foot. 
1  cu.  meter  =  1.308  cu.  yard. 
1  ster  =  .2759  of  a  cord. 


Measures  of  Capacity. 


1  liquid  quart  =  .9463  of  a  liter. 

1  dry  quart  =  1.101  liter. 

1  liquid  gallon  =  .3785  of  a  dekaliter. 

1  peck  =  .881  of  a  dekaliter. 

1  bushel  =  .252-4  of  a  hektoliter. 


1  liter  =  1.0567  liquid  quarts. 

1  liter  =  ,  908  of  a  dry  quart. 

1  dekaliter  =  2.6417  liquid  gallons. 

1  dekaliter  =  1.135  pecks. 

1  hektoliter  =  2.8375  bushels. 


Measures  of  Weight. 


1  grain,  Troy  —  .0648  of  a  gram. 

1  ounce,  Avoir.  =  28.35  gram. 

1  ounce,  Troy  =  31.104gi-ams. 

1  pound,  Avoir.  =  .4536  of  a  kilogram. 

1  pound,  Troy  =  .3732  of  a  kilogram. 

1  ton  (short)  =  .9072  of  a  tonneau. 


1  gi-am  =  .03527  of  an  ounce,  Avoir. 
1  gram  =  .03215  of  an  ounce,  Troy^ 
1  gram  =  15.432  grains,  Troy. 
1  kilogram  =  2.2046  pounds.  Avoir. 
1  kilogram  =  2.679  pounds,  Troy. 
1  tonneau  =  1.1023  tons  (short). 


Remark. — Metric  quantities  of  any  unit  are  read  like  ordinary  decimals. 

I^RENCH    MONEY. 

498.  The  Legal  Currency  of  France  is  decimal,  its  unit  being  the  .sj/t-er 
Franc. 

499,  The  French  coins  are  as  follows: 


Gold 


f  100  francs, 

40  francs, 

20  francs, 

10  francs, 

5  francs. 


(  5  francs, 
-/  2 


Silver  J  2  francs 
/  1  franc. 


Bronze 


f  10  centimes, 
5  centimes, 
2  centimes, 

[   1  centime. 


10  millimes  (m. ) 
10  centimes 
10  decimes 


Table. 

=  1  centime  (ct.)  =  $.00193. 

=  1  dccime  (dc.)  =    .0193. 

=  1  Franc  (fr.)  =     .193. 


MONEY  OF  THE  GERMAN  EMPIRE. 

500.     Tlie  I'nit  is  the  Mark  =  $.2885  United  States  money.     It  is  divided 
into  100  pfennigs  (pennies). 

The  silver  Thaler  =  $.  746  United  States  monev. 


501.     The  German  coins  are: 

(  20  marks. 
Gold  \  10  marks.  Silver 


20  marks, 

10  marks, 

5  marks. 


f  20  marks, 
■j    1  mark, 
(  20  pfennigs. 


Nickel  P?!t""'S«' 
(    0  piejinigs. 


EXAMPLES   IX   DENOMIKATE   NUMBERS.  157 

MISCELI.ANEOUS  KXAMPLES. 

502.  1.  What  is  the  value,  in  English  money,  of  $1750  in  United  States 
gold  coin  ? 

2.  It  required  12  yr.  C  mo.  1  da.  to  build  the  Brooklyn  bridge.  If  it  wjis 
completed  July  4,  1882,  when  Avas  its  construction  begun  ? 

3.  What  is  the  board  measure  of  7  planks,  each  16  ft.  long,  15  in.  wide,  and 
3  in.  thick  ? 

J^.     How  many  acres  of  land  can  be  bought  for  $25000,  if  a  square  foot  costs  25^? 

5.  A  cellar  is  24  ft.  square  inside  of  the  wall,  which  is  9  ft.  high,  and  2  ft. 
thick.     How  tnany  perches  of  IG^  cu.  ft.  each  does  the  wall  contain? 

Remark. — Sometimes  24}  cubic  feet  are  reckoned  as  a  perch,  but  this  is  rarely  done  by 
contractors  or  architects;  girt  measurements  are  taken. 

6.  How  many  shingles,  4  inches  wide,  laid  0  inches  to  the  weather,  would  be 
required  to  cover  the  roof  of  a  barn  GO  ft.  long  and  24  ft.  wide  on  each  side? 

7.  The  highest  chimney  in  the  world  is  at  Port  Dundas,  Scotland,  it  being 
450  ft.  high.     How  many  rods  in  hight  is  it  ? 

8.  The  Italian  Government  pays  out  yearly  $2140000  to  32590  monks  and 
nuns.     What 'is  the  average  sum  received  by  each  ? 

9.  What  will  be  the  cost  of  the  plank,  at  $18  per  M,  that  will  cover  a  floor 
24  ft.  by  13  ft.,  if  the  plank  is  2^  inches  in  thickness  ? 

10.  A  farm  having  225  rods  fronting  the  road,  is  95  rods  wide  atone  end  and 
72.5  rods  at  the  other.     How  many  acres  does  the  farm  contain  ? 

11.  If  the  capacity  of  a  cask  is  64^  wine  gallons,  how  many  quarts  of  berries 
will  it  hold  ? 

12.  A  bird  can  fly  1°  in  1  hr.  10  m.  12  sec.  At  that  rate,  in  Avhat  time  can 
it  encircle  the  earth  ? 

13.  What  will  be  the  cost  in  Paris  of  a  cargo  of  38500  bu.  United  States 
wheat,  at  10  fr.  60  cent,  per  hektoliter? 

14.  How  many  francs  are  equal  to  $275. 

15.  The  largest  shipping  lock  in  the  world  is  at  Cardiff,  it  being  600  ft.  long, 
80  ft.  wide,  and  32  ft.  deep.     What  is  its  capacity  in  barrels  ? 

16.  When  it  is  noon  at  the  point  of  your  observation,  what  is  the  time  at  a 
point  1500  statute  miles  due  south-west  ? 

17.  If  your  coal  costs  $5.  GO  i)er  ton,  and  you  use  G5  lb.  i)cr  day,  wliat  will 
be  the  expense  of  your  fire  for  the  months  of  the  winter  of  1891-2  ? 

18.  How  many  barrels  of  Avater  in  a  cistern  12.5  ft.  long,  10  ft.  wide,  and 
7.5  ft.  deep  ? 

19.  A  carriage  wheel  4  ft.  3  in.  in  diameter  Avill  make  how  many  revolutions 
in  going  62.5  miles  ? 

20.  If  Wm.  II.  Vanderbilt  died  Avorth  two  hundred  millions  of  dollars,  in 
what  length  of  time  could  his  fortune,  in  silver  dollars,  bo  counted  by  one  person, 
counting  GO  per  minute  and  Avorking  10  hours  i)cr  day  for  3G5  days  each  year  ? 

21.  What  will  be  the  cost  of  10  sticks  2  in.  by  4  in.,  10  sticks  2  in.  by  6  in., 
10  sticks  4  in.  by  4  in.,  and  10  sticks  2  in.  by  10  in.,  if  the  sticks  are  each  16  ft. 
long  and  the  cost  is  $15  per  M  ? 


158  EXAMPLES    IN    DENOMINATE    NUMBERS. 

22.  How  many  yards  of  Axminster  carpeting,  f  of  a  yard  in  width,  and  laid 
lengthwise  of  the  room,  will  be  required  to  cover  a  floor  21 J  ft.  long  and  18f  ft. 
wide,  making  no  allowance  for  waste  in  matching  the  design  ? 

23.  How  many  tons  of  324  cu.  ft.  each,  in  a  mow  of  hay  36  ft.  3  in.  long,  18 
ft.  10  in.  wide,  and  13  ft.  6  in.  high  ? 

2^.  Two  astronomers,  located  at  different  points,  observed  at  the  same  instant 
of  time  an  eclipse  of  the  moon,  one  seeing  it  five  minutes  after  9  p.  m.,  local 
time,  and  the  other  five  minutes  before  midnight.  How  many  degrees  of  longi- 
tude separated  the  observers  ? 

25.  How  many  Avoirdupois  pounds  in  10  myriagrams  4  kilograms. 

26.  If  the  sun  is  93  millions  of  miles  from  the  earth,  and  a  cannon  ball  travels 
nine  miles  per  minute,  at  what  time  would  a  ball  fired  from  the  earth  at  one 
minute  after  3  o'clock  p.  m.,  Dec.  25,  1889,  reach  the  sun  at  that  rate  ? 

27.  How  many  francs  are  equal  to  £425  ? 

28.  Allowing  305  sq.  ft.  for  doors  and  windows,  what  will  be  the  cost,  at  40^ 
per  square  yard,  of  plastering  the  ceiling  and  walls  of  a  room  45  ft.  long,  354-  ft. 
wide,  and  12  ft.  3  in.  high  ? 

29.  How  many  German  marks  are  equal  to  $1500  United  States  money  ? 

30.  A  pile  of  wood  built  10  ft.  high  and  22  ft.  wide  must  be  how  long  to 
contain  125  cd.  ? 

31.  The  main  centennial  building  at  Philadelphia  in  18T6  was  1880  ft.  long 
and  464  ft.  wide.     What  was  its  area  in  acres,  square  rods,  and  square  feet  ? 

32.  Reduce  47  mi.  216  rd.  11  ft.  5  in.  to  metric  units. 

33.  If  £2  4  s.  6  d.  is  paid  for  a  coat  and  vest,  and  the  coat  costs  4  s.  more 
than  twice  as  much  as  the  vest,  what  is  the  cost  of  each,  in  United  States  money? 

3^.  From  .001  of  a  section,  plus  .01  of  an  acre,  take  .001  of  a  quarter  section, 
plus  .01  of  a  square  rod. 

35.  A  grocer  bought  12  bu.  of  chestnuts,  at  §3.50  per  bushel  dry  measure, 
and  sold  them  at  Ibf  per  quart  liquid  measure.  Did  he  gain  or  lose,  and  how 
much  ?' 

36.  How  many  dollars  are  equal  to  2150  francs  ? 

37.  How  many  square  feet  of  sheet  lead  will  be  required  to  line  a  tank  7  ft. 
in  diameter  and  12  ft.  deep  ? 

38.  If  bricks  cost  15.50  per  M,  what  will  be  the  cost  of  the  In'ick  for  a  wall 
12  ft.  high  and  3  ft.  thick,  enclosing  an  acre  of  land  10  rd.  wide  and  16  rd.  long  ? 

39.  The  gold  coin  of  the  commercial  world  suffers  each  year  a  loss  of  one  ton 
by  wear  or  abrasion.  What  is  tlie  value,  in  United  States  gold  dollars,  of  the  loss 
thus  sustained? 

JfO.     Reduce  250  hektars  to  common  units. 

41.  What  will  be  the  cost,  at  $16.00  i)er  M,  of  a  tapering  board  18  ft.  long, 
and  9  in.  wide  at  one  end  and  16^  in.  wide  at  the  other  ? 

Jf^i.  A  German  immigrant  having  1000  thalcrs  and  500  marks,  exchanges  them 
for  United  States  money.     How  many  dollars  should  he  receive  ? 

Jf3.  The  hight,  width,  and  length  of  a  shed  are  equal.  What  are  its  dimen- 
sions, if  it  will  contain  125  cords  of  wood  ? 


EXAMPLES   IN    DENOMINATE    NUMBERS.  J 59 

J^.     A  train  of  45  cars  of  Lehigh  coal  averages,  by  the  long  ton,  2o  T.  7  cwt 
3  qr.  to  lb.  per  car.     What  is  the  value  of  the  coal,  at  #5.25  per  short  ton  ? 

Jf5.  How  many  feet  of  lumber  in  a  box  6 J  ft.  long,  h\  ft.  wide,  and  3i  ft. 
deep,  inside  measurements  given,  and  lumber  1  inch  in  tliickness  ? 

J^6.  What  Avill  be  the  cost  of  carpeting  |  yd.  wide,  and  lining  \  yd.  wide,  to 
cover  a  room  24  ft.  long  and  20  ft.  wide,  if  the  strips  of  carpet  are  laid  the 
long  way  of  the  room  and  there  is  a  waste  of  9  inches  at  one  end  in  matching, 
also  an  allowance  of  \^io  in  width  and  %ic  in  length  for  shrinkage  of  the  lining, 
the  carpet  selling  at  $2.25  per  yd.,  and  the  lining  at  30^  per  yd. 

47.  A  pile  of  wood  56  meters  long,  18^  meters  wide,  and  3|  meters  high,  was 
sold  at  $6  per  cord.     How  much  was  received  for  it  ? 

U8.  A  farmer  filled  a  bin  9  ft.  Avide,  12  ft.  long,  and  7^  ft.  deep,  with  wheat 
grown  from  a  field  yielding  32^  bu.  per  acre.  How  long  was  the  field,  if  its 
width  was  50  rods  ? 

J^9. .  Seasoned  pine  in  freighting  is  estimated  to  weigh  3000  lb.  per  M,  and 
green  oak  5000  lb.  per  M.  How  much  freight  must  I  pay,  at  81  i)cr  ton,  on 
a  car  load  of  3205  ft.  of  pine  and  3795  ft.  of  oak  ? 

50.  How  long  is  the  side  of  the  largest  cube  that  can  be  cut  from  a  spherical 
snow  ball  5  ft.  in  diameter  ? 

51.  Glenn's  California  reaper  will  in  12  hours  cut,  thresh,  winnow,  and  put 
into  bags,  30  A.  of  wheat.  How  many  days,  of  15  working  hours  each,  will  it 
require  to  harvest  and  thresh  the  wheat  of  a  field  125  rods  wide  and  240  rods 
long? 

52.  An  ounce  of  gold  can  be  so  beaten  as  to  cover  146  sq.  ft.  What  weight 
of  gold  would  be  required  for  a  sheet  which  will  cover  an  acre  of  ground  ? 

53.  A  farmer  having  1240  bu.  of  corn  in  the  ear  to  store  in  two  rail  cribs, 
builds  each  9  ft.  square  on  the  inside.  \i  one  is  built  10  ft.  high  and  filled,  how 
high  must  the  other  be  built  to  hold  the  remainder  "^ 

5Jf.  The  Hercules  ditcher,  of  Michigan,  removes  750  cu.  yd.  of  earth  per 
hour.  In  how  many  days,  of  12  working  hours  each,  can  it  dig  a  ditch  7  miles 
in  length,  8  ft.  in  deptli,  24  ft.  wide  at  the  surface,  and  10  ft.  at  the  bottom  ? 

55.  If  a  car  carrying  20  tons  of  freight  is  with  its  couplings  42  ft.  long,  what 
would  be  tlie  length  of  a  train  carrying  Vanderbilt's  two  iuindred  millions  of 
dollars,  if  it  is  all  in  standard  silver  dollars,  and  any  fractional  part  of  a  car  load 
18  rejected  ? 


iOO  PERCENTAGE. 


PERCENTAGE. 

503.  Percentage  is  a  term  ajiplied  to  computing  by  hundredths. 

504.  The  Elements  of  Percentage  are,  the  Base,  the  Rate,  the  Amount  Per 
Cent.,  the  Difference  Per  Cent.,  the  Percentage,  the  Amount,  and  the  Difference. 

505.  The  Base  is  the  number  ujion  which  the  percentage  is  computed, 

506.  The  Rate  Per  Cent,  denotes  how  many  hundredths  of  the  base  are  to 
be  taken,  and  is  usually  expressed  as  a  decimal. 

507.  Per  Cent,  is  an  abbreviation  of  the  Latin  words  jt?er  centum,  signifying 
by  the  liundred,  or  a  certain  number  of  each  one  hundred  parts. 

508.  The  Sign,  i,  is  used  to  denote  per  cent. 

509.  The  Rate  may  be  expressed  as  a  part  in  a  common  fractional  form, 
as  f  ;  in  the  form  of  an  extended  decimal,  as  .01625  =  If^  ;  but  only  when 
expressed  in  hundredths  can  it  with  strict  propriety  be  considered  a  rate  per 
cent.     Thus,  .12,  .06,  .15^,  .05f,  are  each  a  rate  per  cent. 

510.  To  read  per  cent. ,  call  the  first  two  places  j!?er  cew^. ,  and  the  added  places, 
if  anv,  fractions  of  1  per  cent.;  as,  .2125  read  as  21  and  one-fourth  per  cent. 

511.  To  express  per  cent,  as  a  common  fraction,  write  the  per  cent,  for  a 
numerator  and  100  for  a  denominator,  and  reduce;  thus,  25^  =  -^^  =  ^. 

512.  To  change  a  common  fraction  to  an  equivalent  per  cent.,  apply  the 
decimal  explanation.  Art.  245.  Divide  the  numerator  by  the  denominator,  and 
give  the  quotient  at  least  two  decimal  places. 

513.  Every  rate  per  cent.,  being  as  many  hundredths,  requires  at  least  two 
decimals  places;  hence,  if  the  per  cent,  be  less  than  10,  a  cipher  must  be  prefixed 
to  the  figure  denoting  it;  thus,  2^  =  .02. 

514.  The  Amount  Per  Cent,  is  100  per  cent,  increased  by  the  rate,  or  1 
phis  the  rate. 

515.  The  Difference  Per  Cent,  is  100  percent,  dimmished  by  the  ra^e, 
or  1  minus  the  rate. 

Remark. — Where  the  rate  per  cent,  is  the  equivalent  of  a  common  fraction,  use  in  solution 
whichever  is  most  convenient. 

516.  The  Percentage  is  the  sum  obtained  by  multiplying  the  base  by  the  rate. 

517.  The  Amount  is  the  sum  of  the  base  and  percentage. 

518.  The  Difference  is  the  remainder  after  deducting  the  percentage  from 
the  base. 


PERCENTAGE.  '  161 

619.  The  Base  is  either  an  abstraot  or  denominate  number;  the  rate  per 
"Cent,  is  always  abstract,  and  the  percentage,  amount,  and  difference  are  always 
like  the  base. 

Remarks.— 1.  In  all  operations  where  a  decimal  rate  is  used,  too  great  care  cannot  be  taken 
to  express  all  decimal  terms  with  exactness. 

2.  As  the  greater  part  of  commercial  calculations  are  based  upon  percentage,  the  importance 
of  a  thorough  mastery  of  its  principles  will  be  readily  perceived. 

520.  ^'\ncQ  per  cent,  is  any  number  of  hundredths,  it  may  be  expressed  either 
as  a  decimal  or  as  a  common  fraction,  and  the  table  of  aliquot  parts  can  be  used 
with  little  variation  and  to  great  advantage  in  many  operations  in  perceyitage. 
Hence,  the  rules  given  under  Special  Applications  may  be  applied  in  this 
^subject. 

Table. 


1\ 


Decimal. 

Com.  Frac.                      Lowest  Terms. 

1 

per  cent. 

= 

.01 

= 

10  0 

reducible  to 

10  0' 

2 

per  cent. 

= 

.02 

= 

10  0 

reducible  to 

t' 

3 

per  cent. 

=z 

.03 

= 

3 

Too" 

reducible  to 

Tinr- 

4 

per  cent. 

= 

.04 

= 

xio- 

reducible  to 

^• 

5 

per  cent. 

= 

.05 

= 

TolT 

reducible  to 

^' 

6 

per  cent. 

= 

.06 

= 

ro*r 

reducible  to 

■io- 

7 

per  cent. 

= 

.07 

= 

tJtt 

reducible  to 

lod* 

8 

per  cent. 

= 

.08 

= 

,  8 
Too 

reducible  to 

ih- 

9 

per  cent. 

1= 

.09 

= 

To¥ 

reducible  to 

106* 

10 

per  cent. 

= 

.10 

= 

Too 

reducible  to 

iV- 

12 

per  cent. 

=: 

.12 

= 

^^ 

reducible  to 

A. 

14 

per  cent. 

= 

.14 

= 

j'^ 

reducible  to 

A- 

16 

per  cent. 

= 

.16 

= 

iVo 

reducible  to 

A. 

20 

per  cent. 

= 

.20 

= 

100 

reducible  to 

i- 

25 

per  cent. 

= 

.25 

= 

tVo 

reducible  to 

i.  • 

30 

per  cent. 

= 

.30 

= 

loo" 

reducible  to 

^. 

50 

per  cent. 

= 

.50 

= 

■^A 

reducible  to 

i. 

75 

per  cent. 

=: 

.75 

= 

100 

reducible  to 

I. 

100 

per  cent. 

= 

1.00 

= 

100 

To  0 

reducible  to 

1. 

125 

per  cent. 

= 

1.25 

= 

12  5. 

1  0  0 

reducible  to 

i  =  u- 

150 

per  cent. 

= 

1.50 

= 

15  0. 

To  0 

reducible  to 

i  =  H' 

li 

per  cent. 

=. 

.0125 

= 

To  0  0  0 

reducible  to 

■sV. 

If 

per  cent. 

= 

.01661 

= 

tIfoot 

reducible  to 

A- 

2i 

per  cent. 

= 

.025 

= 

^2.5_ 

To  0  0 

reducible  to 

iV- 

3i 

per  cent. 

= 

.033^ 

= 

-1  oo_ 
^000 

reducible  to 

A. 

6i 

per  cent. 

= 

.0625 

= 

625 
10000 

reducible  to 

iV- 

8i 

per  cent. 

= 

.0833^ 

= 

2_5_o_0_ 
70000 

reducible  to 

iV. 

m 

per  cent. 

= 

.125 

= 

JL2_5_ 

To  0  0 

reducible  to 

I 

m 

per  cent. 

z= 

.1661 

= 

WW 

reducible  to 

h 

33^ 

per  cent. 

= 

.333  J 

= 

"To  0  0 

reducible  to 

h 

62^ 

per  cent. 

= 

.625 

=. 

To  0  0 

reducible  to 

l. 

66f 

per  cent. 

= 

.661 

=z 

m 

reducible  to 

f. 

m 

per  cent. 

= 

.875 

= 

A^oV 

reducible  to 

h 

Ifi2  EXAMPLES   IN    PERCENTAGE. 

521.  Tlie  ri'latiou  between  the  eleinents  of  Percentage  is  such,  tliat  by  the- 
application  of  the  General  Principles  of  MultiiiJieation  and  Division,  if  any 
two  of  the  elements,  except  amount  ]ier  cent,  and  difference  per  cent.,  are 
given,  the  other  three  may  be  found. 

522.  To  find  the  Percentage,  the  Base  and  Rate  being  given. 

Exam PLK.— What  is  25^  of  1:468  ? 

First  Explaxatiox. — 25  per  cent,  equals  .25;  therefore, 

Operation.  ^5  per  cent,  of  $468  equals  $468  multiplied  by  .25,  equals 

$468  =  base.  $117. 

,25  =  rate  per  cent.  Second  Explanation. — $468  is  100  per  cent,  of  itself; 

r~~zr7T  ^  and  since  25  ner  cent,  equals  i  of  100  per  cent.,  25  per  cent. 

$117.00  =  percentage.  ^^  ^^^^  ^,^  -^  ^  ^^  ^^^^  ^^^  ^^  ^^^^ 

Rules. — 1.    Multiply  the  base  by  the  rate  expressed  decimally.    Or, 
2.    Take  such  a  part  of  the  base  as  the  number  expressing  the  rate  is 

part  of  1. 

Remark. — When  the  rate  is  an  aliquot  part  of  100,  the  percentage  may  be  found  by  taking 

a  like  part  of  the  base:  thus,  for  10'?  take  iV,  for  ib'i  take  i,  for  33^-;  take  \,  etc. 

Formula. — Percentage  =  Base  X  Rate. 

KXAMPLE.S   FOK  MEIfTAI.  PKACTICE. 


523.  What  is 

1.  5  per  cent,  of  100  ? 

2.  12  per  cent,  of  600  ? 

3.  15  per  cent,  of  800  ? 


J^.     20  per  cent,  of  500  ? 

5.  25  per  cent,  of  1200  ? 

6.  33^  p^r  cent,  of  -^^  ? 


7.  25  per  cent,  of  1440  ? 

8.  8  per  cent,  of  450  ? 

9.  50  per  cent,  of  680  ? 


examples  for  written  practice. 

524.     1.     A  man  owning  250  acres  of  land,  sold  20^^  at  one  time,  and  25^  of 
the  remainder  at  another  time.     How  many  acres  did  he  have  left  ? 

2.     If  a  ranchman  having  5450  sheep,  lost  20^^  by  a  storm  and  'afterwards 
sold  20*^  of  those  remaining,  how  many  sheep  did  he  sell? 

5.  A  collector  deposited  813500  in  coin,  and  12|^  more  in  bank  bills.     What 
was  the  total  of  his  deposit  ? 

J^.     Find  ll^f^  of  1G80  lb.  of  wool. 
J.     Find  1655^  of  12  lb.  3  oz.  of  silver. 

6.  From  a  charge  of  $675,  made  for  a  bill  of  goods,  8j^  was  deducted.  What 
was  the  net  amount  of  the  bill  ? 

7.  If  526  barrels  of  salt  were  bought  for  $1.10  per  bar. ,  and  sold  at  an  advance 
of  15^,  what  was  gained  ? 

8.  Two  men,  each  having  $12500,  made  investments,  from  which  one  gained 
15j^,  and  the  other  lost  35^^.     How  much  did  each  then  have? 

9.  How  much  greater  is  12^^  of  $1550,  than  74^  of  $2150  ? 

10.  Having  raised  1240  bushels  of  wheat,  a  farmer  used  5^  of  it  for  seed  and 
5^  for  bread;  he  then  sold  to  one  man  10^  and  to  another  25^  of  what  remained. 
How  many  bushels  had  he* left  ? 


EXAMPLES    IN    PERCEHTTAGE.  163 

11.  Having  $75000  to  invest,  u  gentleman  bought  United  States  bonds  with 
3iH^  of  his  money,  a  home  with  20^^,  and  invested  the  remainder  equally  in  farm 
lands  and  manufacturing  stock.     How  much  did  he  pay  for  tlie  farm  lands  ? 

12.  I  owed  John  Smith  $1750,  and  paid  at  one  time  'HOfo  of  the  debt,  at 
another  time  35^  of  the  remainder,  and  at  another  time  25^  of  what  then  re- 
mained unpaid.     How  much  of  the  debt  did  I  still  owe  ? 

13.  A  capitalist  owning  |  of  a  coal  mine,  sold  324^  of  his  share  for  $65000. 
At  that  rate,  what  was  the  entire  mine  worth  ? 

IJf..  A  jobber  having  bought  2160  bags  of  coffee,  sold  at  one  time  8^^,  at 
another  25^  of  what  remained,  and  at  a  third  sale  15^  of  what  still  remained. 
Find  the  value  of  what  was  left,  at  $18  per  bag. 

15.  Of  a  farm  containing  a  half,  section  of  land,  15^  was  in  Avheat,  32^  in 
oats,  5^  in  potatoes,  and  the  remainder  devoted  equally  to  orchard,  corn,  beans, 
and  pasture.     How  many  acres  were  in  pasture  ? 

16.  A  farmer  having  156  sheep  to  shear,  agreed  to  pay  for  their  shearing  4^ 
of  the  sum  received  for  their  wool.  If  the  fleeces  averaged  74-  lb.  and  sold  for 
30^  per  pound,  how  much  was  paid  for  shearing  ? 

17.  A  speculator  having  $41820,  invested  50^'  of  it  in  oil,  on  which  he  lost 
IQ'^fc  ;  the  remainder  he  invested  in  cotton,  which  he  sold  at  9^  below  cost. 
How  much  was  received  from  both  sales  ? 

18.  A  trader  bought  12  mustangs  for  $400,  and  after  selling  2b%  of  the  num- 
ber at  a  gain  of  50^,  and  33^^^  of  those  remaining  at  a  gain  of  12^^,  sold  those 
still  on  hand  at  $30  per  head.     Did  he  gain  or  lose,  and  how  much  ? 

525.     To  find  the  Base,  the  Percentage  and  Rate  being  given. 

Remark. — Since  the  base  multiplied  by  the  rate  produces  the  percentage,  percentage  must 
be  a  product;  if,  therefore,  it  is  divided  by  either  factor,  the  quotient  will  be  the  other  factor. 

Example. — By  selling  4^  of  a  stock  of  goods,  a  merchant  realized  $644. 
What  was  the  value  of  the  entire  stock  ? 

Operation. 
Rate.    Percentage.  Explanation.— If  the  value  of  4  per  cent,  is  $644,  the  value  of 

.04  )  644.00  ^  P^'"  ^^°*-  ^^^^  ^^  %IQI\  and  if  the  value  of  1  per  cent,  is  $161,  the 

'- value  of  100  per  cent,  will  be  $16100. 

16100  base. 

'R\\\e.— Divide  the  percentage  hy  the  rate,  expressed  decimally. 
Formula.— Base  =  Percentage  -^  Rate. 

KXAMPIiES  FOR  MENTAL,   I'KAOTICE. 

626.     1.     846  =  Q</c  of  what  number  ? 

2.  2150  =  10^  of  what  number  ? 

3.  543  =  b'/o  of  what  ntimber  ? 
1     219  =  33^^  of  what  number  ? 

5.  150  =  ^^0  of  what  number  ? 

6.  A  man  sold  25;^  of  his  farm  for  $2120.  How  luut-li  was  the  farm  worth 
at  that  rate  ?  » 


164  EXAMPLES   IN    PERCENTAGE. 

7.  What  is  the  value  of  a  liouse  renting  for  $300  per  year,  if  the  rent  equals 
9^  of  its  value  ? 

8.  HoAv  many  acres  in  a  farm  of  whicli  12.5  acres  is  but  5^. 

9.  Of  what  sum  is  136  but  33^^  ? 

EXAMPLES   FOR   W'RITTEX   PRACTICE. 

527.  1.  A  planter  sold  76  bales  of  cotton,  which  was  19^  of  his  crop.  How 
many  bales  did  he  raise  ?  ^ 

2.  I  paid  $123.48,  which  was  16|'^  of  a  debt.     What  amount  did  I  owe  ? 

3.  A  lady  i)aid  for  millinery,  $17.50;  for  shoes,  $11.40;  for  jewelry,  113.80; 
for  furs,  $78.55;  and  had  expended  but  15^  of  her  money.  How  many  dollars 
had  she  at  first  ? 

^.  A  clerk's  present  salary  of  $520  per  year  is  only  75'^  of  what  he  formerly 
received.     How  much  was  formerly  paid  him  ? 

5.  A  grocer,  after  increasing  his  stock  to  the  amount  of  $6448,  found  that 
the  new  purchase  was  but  16^  of  the  old  stock  on  hand.  What  was  tlie  value 
of  his  old  stock  ? 

6.  The  owner  of  68^  of  a  mine,  received  $91510  from  the  sale  of  25^  of  his 
share.     Find  the  value  of  the  entire  mine  at  that  rate  ? 

7.  A,  B,  C,  and  D  are  partners;  A  furnished  15,'^  of  the  capital,  B  25ji^,  C 
42f;^,  and  D  $16200.     What  was  the  capital  of  the  firm  ? 

8.  A  Wyoming  ranchman  lost  1684  cattle  during  a  blizzard.  Hom'  many  had 
he  at  first,  if  his  loss  was  only  If^  of  his  herd  ? 

9.  The  population  of  a  county  increased  22j^  in  ten  years.  If  the  births 
exceeded  the  deaths  by  2166,  and  the  county  received  13234  immigrants  during 
the  time,  what  must  have  been  its  jwpulation  before  the  increase  ? 

10.  A  speculator  owned  a  quarter  interest  in  a  mill,  and  sold  one-quarter  of 
his  part  for  $11250.     What  was  the  mill  worth,  on  that  basis  of  value  ? 

528.  To  find  the  Rate,  the  Percentage  and  Base  being  given. 

Remark. — The  percentage  is  a  product,  the  base  being  one  of  its  factors. 

Example.— What  per  cent,  of  480  is  120  ? 

First  Operation. 

4.80  )  120.00  (  25  times.        Ftrbt  Explanation.— Since  480  is  100  per  cent,  of  itself,  1 

960  per  cent,  of  480  would  be  ^l^  part  of  it,  or  4.80;  and  since  4.80 

is  1  per  cent,  of  480,  120  would  be  as  many  times  T  per  cent. 

as  4.80  is  contained  times  in  120,  which  is  25  times;  and  25 

times  1  per  cent.  =  25  per  cent. 


2400 

1^ -.01  2400 

25 


.25  =  25^1^. 


Second  Operation.  „            ^                        «.         ,                      .            ^ 

Aftfi  "\  190  ftO  /"    9"  —  'i^'i  Second  Explanation. — Smce  the  percentage  is  a  product 

qA()               ~     '  '  *  of  the  base  and  rate,  the  quotient  obtained  by  dividing  the  per- 

ceptage  by  the  base  will  be  the  rate.     Or,  120  is  ||8,  or  J 

2400  of  480;  and  since  480  is  100  per  cent,  of  itself,  120,  which  is 

2400  J  of  480,  must  be  i  of  100  per  cent.,  or  25  per  cent. 


EXAMPLES   I?f   PERCENTAGE.  I<i5 

^ule.—Dii/ide  the  percentage  by  the  base,  carrying  the  quotient  to  two 
decimal  places. 

Formula.  —Rate  =  Percentage  -i-  Base. 

EXAMPI.KS   FOK   MENTAIv   PRACTICE. 

529.     What  per  cent,  is 
1. 


25  of  125  ? 

I 

12i  of  100  ? 

7. 

37^  of  150  ? 

40  of  160  ? 

5. 

15  of  45  ? 

8. 

200  of  10  ? 

18  of  36  ? 

6\ 

125  of  1000  ? 

9. 

120  of  4  ? 

EXAMPLES  FOR  WRITTEN   PRACTICE. 

530.  i.    From  a  herd  of  1184  cattle,  296  were  sold.     "What  per  cent,  was  sold? 

2.  R.  G.  Dun  &  Co.  charged  821  for  collecting  an  account  of  1600.  What 
rate  was  charged  ?  ' 

3.  Sold  f  of  a  stock  of  goods  for  what  the  entire  stock  cost.  What  was  my 
rate  of  gain  ? 

^.     What  per  cent,  of  12  lb.  8  oz.  is  2  lb.  8  oz.,  Avoirdupois  ? 

5.  From  a  half  section,  120  acres  were  sold,  and  afterwards  80  acres  more. 
What  per  cent,  was  sold  ? 

6.  Of  a  stock  of  800  bushels  of  potatoes,  240  bushels  were  sold  at  one  time, 
and  135  bushels  at  another.     What  per  cent,  was  still  unsold  ? 

7.  A  merchant  failed,  owing  $27984,  liis  assets  amounting  to  l>16090.80. 
What  per  cent,  of  his  debts  can  he  pay  ? 

8.  At  a  normal  school  there  were  enrolled  855  male  pupils  and  only  185  female 
pupils.     What  per  cent,  more  were  the  male  than  the  female  pupils  ? 

9.  A  girl  having  $5.40,  expended  11.35  for  gloves,  45^  for  flowers,  and  one- 
half  of  the  remainder  for  a  pair  of  slippers.  What  per  cent,  of  her  money  had 
she  left  ? 

10.  From  a  cask  of  lard  of  314  lb.,  78.5  lb.  were  sold  at  one  time,  and  25^  of 
the  remainder  at  another.     What  per  cent,  of  the  whole  remained  unsold  ? 

11.  Of  a  regiment  of  men  entering  battle,  1040  strong,  only  260  came  out 
unhurt,  ^  of  the  remainder  having  been  killed.  What  per  cent,  of  the  whole  were 
killed  ? 

531.  To  find  the  Amount  Per  Cent.,  the  Rate  being  given. 

Example. — If  the  rate  be  7^,  what  is  the  amount  per  cent.  ? 

Operation  Explanation. — Since   the  amount  per   cent. 

-iQQ^  1         _       unit  (definition,  page  160),  is  always  100  per  cent,  in- 

'             '  creased  by  the  rate,  we  may  tind   it  by  adding 

7%  =    .  0  (  =  rate  jqq  p^^  ^^^^  ,  or  1 ,  to  the  per  cent,  given.     Hence, 

1  A.>v       ~rr^r^  J.  ,         if  the  rate  is  7  per  cent.,  the  amount  per  cent,  will 

107^  =:  1.07  =  amount  per  cent.       .    .^-  ^ 

^  be  107  per  cent. 

Rule.— ^dc?  the  rate  to  the  unit  1. 

Formula. — Amount  Per  Cent.  =  1  4-  Rate. 


106  EXAMPLES   IX   PERCENTAGE. 

EXAMPLKS   FOR  MENTAL,   PRACTICE. 

53*2.     1.     If  the  rate  be  10;e,  ■what  will  be  the  amount  per  cent.? 

2.  If  the  rate  be  75;;^,  what  will  be  the  amount  per  cent.  ? 

3.  If  the  rate  be  110'^,  what  will  be  the  amount  i)er  cent.? 
Jf.  Find  the  amount  per  cent.,  if  the  rate  per  cent,  be  I65? 
<5.     Find  the  amount  per  cent.,  if  the  rate  per  cent,  be  8T^  ? 

EXAMPLES   FOR   WRITTEN'   PRACTICE. 

533.  1.  Goods  costing  1*14:00  were  sold  for  $14T0.  Find  the  amount  per 
cent,  of  the  selling  price  ? 

2.  Last  month  I  sold  $"2750  worth  of  coffee,  while  the  previous  month  I  sold 
$3000  worth.  What  was  the  amount  per  cent,  of  my  sales  for  tlie  previous  month 
as  compared  with  those  of  the  last  month  ? 

S.  If  tea  costing  ^'2\^  per  pound  sell  at  87^^,  what  amount  per  cent,  do  the 
sales  show  as  compared  with  tlie  cost  ? 

534.  To  find  the  Difference  Per  Cent.,  the  Rate  being  given. 
Example. — If  the  rate  be  b^,  what  is  the  difference  per  cent.? 

Operation.  Explaxatiox. — Since  the  difference   per  cent,   (definition, 

100^  =  1.00  =  a  tinit.  p^gg  jgo),  is  equal  to  100  per  cent.,  or  1,  less  the  rate,  if  we  take 

0  <r  =     .Oo  =  rate.  the  given  rate,  5  per  cent.,  from  100  per  cent.,  the  remainder, 

95^'  --      C)^  -—  (Jif_  c^^  95  per  cent.,  will  be  the  answer  required. 

Rule. — Subtract  the  rate  frow  the  unit  1. 

Formula. — Difference  Per  Cent.  =  1  —  Rate. 

EXA3IPI.ES   FOR    3IEXTAL    PRACTICE. 

535.  1.     If  the  rate  be  lofc,  what  is  the  difference  i)er  cent.  ? 

2.  If  the  rate  be  37^^,  what  is  the  difference  per  cent.  ? 

3.  If  the  rate  be  \'i,  what  is  the  difference  per  cent.  ? 
-^  If  the  rate  be  3^^,  what  is  the  difference  per  cent.  ? 
5.  If  the  rate  be  70f^,  what  is  the  difference  per  cent.  ? 

EXAMPLES  FOR   AVRITTEX  PRACTICE. 

536.  1.  The  pujiils  of  a  school  are  reduced  in  number  from  112  to  80. 
What  per  cent,  is  the  ])rescnt  of  the  former  attendance  ? 

2.  Walter,  having  48  marbles,  gave  Henry  15.     What  per  cent,  liad  lie  left  ? 

3.  Find  the  difference  per  cent.,  if  tlie  rate  equals  -J  of  ^. 

537.  To  find  the  Amount,  the  Base  and  Rate  being  given. 
Example. — Wliat  i.s  the  amount  of  r).j(i  increased  l>y  8<  of  itself? 

^'  Explanation. — The  amount  equals  base  plus  percentage  (de- 

550  =  base.  finition,  page  160).     The  base  is  550  and  8  per  cent,  of  550  equals 

•^"        rate.  4^^  ^jj^  percentage;  therefore  the  amount  niuist  equal  550  plus  44, 

44.00  =  per   cent,  or  594;  or,  .'Jince  550  equals  100  per  cent,  of  itself,  an  increase  of  8 

550        =  base.  per  cent,  would  give  108  per  cent,  of  the  original  number;  and 

108  per  cent,  of  550,  uv  1.08  times  550  equals  594. 

594        =  amount 


EXAMPLES    IX    PEKCENTAGE.  167 

Rules.— i.    Find  the  -percentage  and  add  it  to  the  base.    Or, 
^.    Multiply  the  base  by  1  plus  the  rate- 

Formula. — Amount  ==  Base  +  Percentage. 

EXAMPLES  FOR  :»IEXTA1.  PKACTICE. 

638.  i.     If  the  base  is  1500,  and  the  rate  10^  what  is  the  amount  ? 
2.     If  the  base  is  1356,  and  the  rate  25^,  what  is  the  amount  ? 

S.  The  base  is  440  and  the  rate  5^;  find  the  amount. 

Jf..  The  base  is  1000  and  the  rate  18^;  find  the  amount. 

5.  The  base  is  252  and  the  rate  10^;  find  the  amount. 

6.  The  base  is  2150  and  the  rate  20^;  find  the  amount. 

7.  The  base  is  630  and  the  rate  33|^;  find  the  amount. 

8.  The  base  is  546  and  the  rate  16|^,*  find  the  amount. 

9.  The  base  is  200  and  the  rate  125^;  find  the  amount. 

EXAMPr.ES  FOR  WKITTEX   PRACTICE. 

639.  1.     What  amount  will  be  received  for  a  house  costing  %13500,  if  it  is 
sold  at  a  gain  of  l^fo  ? 

2.  A  bought  two  horses  for  $180  each,  and  sold  one  at  a  gain  ef  20^  and  the 
other  at  a  gain  of  33^^^^.     How  much  did  he  receive  for  both  ? 

3.  A  section  of  Kansas  prairie  was  bought  at  $12.50  i)er  acre,  and  sold  at  an 
advance  of  40;^.     How  much  was  received  for  it  ? 

4.  What  is  the  amount  of  768  increased  by  25,^^  of  \  of  itself  ? 

5.  What  is  the  amount  of  $3144  increased  by  f  of  16f  ^  of  itself  ? 

6.  If  the  base  is  $864.88  and  the  rate   3^^   of   f   of    itself,    what    is    the 
amount  ? 

640.  To  find  the  Dijfference,  the  Base  and  Rate  being  given. 

Example. — What  remains  after  diminishing  450  by  10^^  of  itself  ? 
Operation. 

100^  =  450      =  base. 

\Qc^               ^     jQ      _  j.j^te.  Explanation. — Since   100  per  cent,  of  the 

—^ number  equals  450,  10  per  cent,  of  it  will  equal 

90^  dif.   ^        45.00  =  percentage.  45.  and  450  minus  45  equals  405.     Or,  since  100 

450  base  P^^  *^^°*'  ^Q*^^!^  450,  10  per  cent,  less  than  100 

45  percentage  P^*^  cent.,  or  90  per  cent,  will  equal  405. 

405  difference. 

Rules.— i.    Find  the  percentage  and  subtract  it  from  the  base.    Or, 
£.    Multiply  the  base  by  1  minus  the  rate. 

Formula. — Difference  =  Base  —  Percentage. 

EXAMPLES   FOR   3IENTAL   PRACTICE. 

641.  1.     If  from  a  brood  of  15  chickens  20r»  arc  lost,  how  many  will  remain ? 
3.     What  number  will  remain  if  225  is  diminished  by  33^^^  of  itself  ? 

3.     If  the  base  is  1050  and  the  rate  10^,  what  is  the  difference  ? 


168  EXAMPLES   IN   PERCENTAGE. 

4.     816,  less  25^  of  itself,  equals  what  number  ? 
0.     1440,  less  165^  of  itself,  equals  what  number  ? 

6.  800,  less  3T^^  of  itself,  equals  what  number  ? 

7.  40,  less  87^;^  of  itself,  equals  what  number  ? 

8.  A  boy  having  648  ft.  of  kite  string,  lost  12|^  of  it.     How  many  feet  had 
he  remaining  ? 

£XAMPI^S   FOK   WKITTEX   PRACTICE. 

542.  ,  1.     A  speculator  lost  35<^  of  ^  of  $16250.     How  much  did  he  lose? 

A?.     A  planter  having  616  acres  in  rice,  lost  \  of  33^^^  of  his  planting  by  flood.. 
How  many  acres  had  he  left  for  harvest  ? 

3.  Brown  deposited  $1147  in  a  savings  bank,  and  his  son  deposited  21,<^  less. 
How  much  was  deposited  by  both  ? 

4.  An  agent  earned  $250  in  May,  15^  less  in  June,  and  20^^  less  in  July  than 
in  June.     What  was  the  amount  earned  for  the  three  months  ? 

543.  To  find  the  Base,  the  Amount,  or  Difference,  and  the  Rate  being  given. 
Example  (first   illustration). — What   number,    increased   by  15^   of  itself,. 

amounts  to  345  ? 

ExPLASATiox. — Since  the  number  must  be  100 

J 00,^  l.«  I  p^j.  pgQj  Qf  itself,  if  it  has  been  increased  15  per 

^^^  =  j}^  Amount.    Base.        (.^^^  ^^^  ^just  be  115  per  cent,  of  that  number; 

llb^  amt.  J^      1.15  )  345.00  (  300       if  115  per  cent,  is  345,  1  per  cent,  must  be  yl^  of 

345  345,  or  3;  and  100  per  cent,  will  be  100  times  3,  or 

■~oo  ^- 

Example  (second  illustration). — What  number,  diminished  by  35^  of  itself^ 

equals  975  ? 

Oper.\.tiox. 

1    r.A 

ExpLAKATiOK. — If  the  number  be  diminished 
by  35  per  cent,  of  itself,  there  will  be  remaining  but 
65  per  cent,  of  itself;  and  if  65  per  cent,  of  the 
number  be  975,  1  per  cent,  must  be  ^V  of  975,  or 
15;  and  if  1  per  cent,  be  15,  100  per  cent,  must  be 
1500. 

00 
Rules.— i.    Divide  the  amount  hij  1  plus  the  rate.    Or, 
2.    Divide  the  difference  hy  1  minus  the  rate. 

Formulas. — 1.  Base  =  Amount  -^  Amount  Per  Cent. 

2.  Base  =  Difference  -=-  Difference  Per  Cent. 

EXAMPLES  FOK  MENTAL   PKACTICE. 

544.*    1.     If  the  amount  is  750  and  the  rate  25'^,  what  is  the  base  ? 

2.  What  number,  increased  by  10'^  of  itself,  amounts  to  440  ? 

3.  After  loi  of  a  number  had  been  added  to  it,  the  amount  was  525.     What 
was  the  number  ? 


Oper.\tiox. 

.00^ 

=  1.00 

35;^ 
65^ 

dif. 

=    .35 

Diff. 

I  975. 
65 

325 
325 

00 

Base. 
(1500 

REVIEW   OF   THE    PRINCIPLES   OF    PERCENTAGE.  16^ 

^.  After  selling  '60^  of  his  apples,  a  boy  had  70  left.  IIow  many  had  he  at 
first  ? 

5.  I  lost  $600  by  a  bankrupt,  who  paid  only  85^  of  his  indebtedness.  What 
was  the  fnll  amount  of  my  claim  ? 

EXAMPLES  FOK   WKITTEN  PRACTICE. 

645.  -?.  A  builder  gained  15^  by  selling  a  house  for  $1150.  What  was  ita 
cost  ? 

2.  Sold  945  tubs  of  butter  for  $5113,  and  thereby  gained  20^.  Ilbw  much 
did  the  butter  cost  per  tub  ? 

3.  The  income  from  a  tenement  house  is  $6042  this  year,  which  is  24^  less 
than  it  Avas  last  year.     How  much  was  it  last  year  ? 

4.  A  liveryman  paid  $180  for  a  horse,  which  was  40^  less  than  he  paid  for  a 
carriage.     How  much  did  he  pay  for  both  ? 

5.  A  drover  gained  16f  ^  on  33  head  of  cattle  sold  for  $4081.  What  was  the 
average  cost  jier  head  ? 

6.  Smith  sold  two  horses  for  $1500  each,  gaining  25^^  on  one,  and  losing  25^ 
on  the  other.     What  did  the  horses  cost  him  ? 

7.  After  paying  35^  of  his  debts,  a  man  finds  that  the  remainder  can  be  paid 
with  $19500.     W^hat  was  his  entire  indebtedness  ? 

8.  A  boat  load  of  wheat  was  so  damaged  that  it  was  sold  for  $8500,  which 
was  15^  less  than  its  original  value.     What  was  its  value  before  it  was  damaged? 

9.  The  attendance  of  pupils  at  a  school  during  May  was  954,  which  was  6^ 
more  than  attended  during  April,  and  this  was  80^  more  than  attended  during^ 
February.     What  Avas  the  attendance  for  Februtiry  ? 

10.  Which  is  better,  to  invest  in  a  house  that  Avill  rent  for  $30  per  month,  at 
6^  on  its  value,  or  to  invest  the  same  amount  in  a  farm  that  in  two  years  will 
bring  $7000  ?     How  much  better  in  the  two  vears  ? 


REVIEW  OF   THE   PRINCIPLES   OF   PERCENTAGE. 

546.  1.  To  find  the  percentage,  the  base  and  rate  being  given.  Kule. — 
Multiply  the  base  ly  the  rate  expressed  decimally. 

2.  To  find  the  base,  the  percentage  and  rate  being  given.  Rule. — Divide 
the  lyercentage  hy  the  rate  expressed  decimally. 

3.  To  find  the  rate,  the  percentage  and  base  being  given.  Rule. — Divide 
the  percentage  hy  the  base,  carrying  the  quotient  to  tivo  decimal  places. 

4.  To  find  the  amount  per  cent.,  the  rate  being  given.  Rule. — Add  the 
rate  to  the  unit  1. 

6.  To  find  the  difference  per  cent.,  the  rate  being  given.  Rule. — Subtract 
the  rate  from  the  unit  1. 

6.  To  find.tlic  amount,  the  base  and  rate  being  given.  Rules. — 1.  Mul- 
tiply the  base  by  the  rate,  and  to  the  product  add  the  base.  Or,  2.  Multiply  the 
base  by  100  per  cent,  plus  the  rate. 


170  EXAMPLES    FOR    PRACTICE    TX    PERCENTAGE. 

7.  To  find  the  difference,  the  base  and  rate  being  given.  Rules. — Multiply 
Ihe  base  by  the  rate,  and  subtrai't  the  product  from  the  base.  Or,  Multiply  the 
base  by  100 per  cent,  minus  the  rate. 

8.  To  find  the  base,  the  amount  and  rate  being  given.  Rule. — Divide  the 
amount  by  100  per  cent,  plus  ihe  rate. 

9.  To  find  the  base,  the  difference  and  rate  being  given.  Rule. — Divide 
the  difference  by  100  per  cent,  minus  the  rate. 

547.  Percentage  is  applied  to  two  chisses  of  problems: 

First,  to  those  in  which  time  is  not  an  element;  as.  Profit  and  Loss,  Com- 
mission, Brokerage,  Insurance,  Taxes,  Customs  or  Duties,  and  Trade  Discounts. 

Second,  to  those  in  which  time  enters  as  an  element;  as.  Interest,  Bank  Dis- 
count, True  Discount,  Equation  of  Accounts,  and  Exchange. 

Remark. — The  pupil  should  be  drilled  in  the  formulas  and  rules  of  simple  or  abstract 
Percentage  as  above,  and  in  their  application  to  problems  in  applied  Percentage  to  follow. 

MISCELLANEOUS   EXABIPLES   FOR  PRACTICE. 

548.  1.  At  the  battle  of  Waterloo,  of  the  145000  combatants,  51000  were 
either  killed  or  wounded.     What  per  cent,  were  uninjured  ? 

2.  The  pressure  on  a  steam  boiler  was  61.2  lb.,  after  it  had  been  reduced  lOj^. 
What  was  it  before  the  reduction  ? 

3.  A  pupil  in  examination  answered  correctly  56  qitestions,  which  was  20^1^ 
less  than  the  number  asked  him.     What  should  be  his  average,  on  a  basis  of  100? 

4.  By  assessing  a  tax  of  f^,  $175000  was  raised  in  a  county.  What  amount 
of  property  was  taxed  ? 

5.  A  benevolent  lady  gave  $10500  to  three  charities;  to  the  first  slie  gave 
$2500,  to  the  second  $4500,  and  to  the  tiiird  the  remainder.  What  per  cent,  did 
each  receive  ? 

6.  On  attaining  his  majority,  a  son  finds  liis  age  is  62-k^  less  than  the  age  of 
his  father.     Find  tlie  sum  of  their  ages  ? 

7.  If  8^  of  B's  money  equals  245^  of  C*s,  how  much  has  C,  if  B  has  $324  ? 
S.     A  farmer  bought  a  horse,  a  mule,  and  a  cow,  for  $385.     The  mule  cost 

15^  less  than  tlie  horse,  and  the  cost  of  the  cow  was  7^^  of  that  of  the  horse. 
What  was  the  cost  of  each  ? 

9.  A  creditor,  after  collecting  21f^  of  a  claim,  lost  the  remainder,  which 
was  $3918.75.     What  was  the  sum  collected  ? 

10.  A  woman  weaving  a  rag  carpet  used  185'^  more  weight  of  rags  than  of 
war}).     How  many  pounds  of  each  in  a  bale  of  carpet  weighing  96^  pounds  ? 

11.  The  sum  paid  for  two  watches  was  $384,  and  75jfe  of  the  sum  paid  for  one 
equalled  105-^  of  the  sum  paid  for  the  other.     Find  the  price  of  each. 

12.  If  Abas  35'^  more  money  than  B,  and  B  has  25f^  more  than  C.  how  much 
has  C,  if  A  has  $102  ? 

13.  If  a  gain  of  $4755  was  taken  out  of  a  business  at  the  end  of  the  first 
year,  and  a  loss  of  $3566.25  was  sustained  tlie  second  year,  what  was  the  per  cent, 
of  net  gain  or  loss,  the  investment  having  been  $63400  ? 


EXAMPLES    FOR    PRACTICE    IX    PERCENTAGE.  171 

14.  After  making  three  of  the  seven  equal  annual  payments  of  the  face  of  a 
mortgage,  I  find  $5850  to  be  still  unpaid.  How  many  dollars  of  principal  have 
been  paid  ? 

15.  After  the  salary  of  a  book-keeper  had  beenincreased.  10^,  and  afterwards 
8^,  he  received  $1242  a  year.     What  Avas  his  salary  at  first  ? 

16.  By  the  United  States  Census  of  1880  the  total  capital  invested  in  man- 
ufactures in  the  State  of  Pennsylvania  was  $190055904,  while  the  amount  invested 
in  Alabama  was  $9098181;  Arkansas,  $131GG10;  Delaware,  $5452887;  Florida, 
$1874125;  Georgia,  $10890875;  Louisiana,  $7151172;  Mississippi,  $4384492; 
North  Carolina,  $9693703;  South  Carolina,  $6931756;  Texas,  $3272450.  What 
per  cent,  greater  was  the  manufacturing  capital  invested  in  Pennsylvania  than  in 
the  group  of  the  ten  other  States  named  ? 

17.  From  an  estate  tlie  widow  received  $9250,  which  was  one-third;  the 
remainder  'Avas  divided  among  three  children,  aged  respectively  15,  12,  and  10 
years,  and  they  shared  in  proportion  to  their  age.  What  per  cent,  of  the  estate 
did  each  of  the  children  receive  ? 

18.  A  herder  was  asked  how  many  cattle  he  had,  and  replied:  "My  herd 
increased  last  year  40^;  should  it  increase  at  the  same  rate  during  this  year  and 
next,  and  I  then  buy  4  head  more,  I  shall  have  double  my  present  number." 
How  many  head  of  cattle  had  he  ? 

19.  What  per  cent,  of  the  amount,  at  10^,  is  10^  of  the  base  ? 

20.  From  a  farm  containing  180  A.  120  sq.  rd.,  one-half  was  sold  at  one  time, 
and  one-half  of  the  remainder  at  another  time.  What  per  cent,  of  the  whole 
then  remained? 

21.  A  man  drew  15f^  of  his  deposit  from  a  bank,  and  with  it  paid  a  debt  of 
$1119.60.     What  balance  was  left  in  the  bank  .? 

22.  Ibfo  of  f  of  a  number  is  wliat  per  cent  of  |  of  it  ? 

2S.  A  man  sold  two  farms  for  $7500  each;  for  one  he  received  25^  more  than 
it  cost,  and  for  the  other  25^  less  than  it  cost.  Did  he  gain  or  lose  by  the 
sale,  and  how  much  ? 

2Jf..  What  number  is  that  which,  being  increased  by  35;^  and  46^  of  itself 
and  76  more,  will  be  doubled  ? 

25.  A  ranchman,  when  asked  how  many  sheep  he  had,  replied:  "  If  my  flock 
increases  next  year  20^,'  the  next  25^^,  and  the  third  year  40^,  I  can  then  sell  300, 
and  have  left  double  my  present  number.     How  many  had  he  ? 

26.  The  total  number  of  Popes  up  to  1888  has  been  253,  of  whom  197  have 
been  Italians.     What  per  cent,  of  all  have  been  of  that  nationality  ? 

27.  By  widening  a  roadway  5j^,  it  was  made  lO.V  yd.  wide.  What  was  its 
original  width  ? 

28.  Oct.  11,  1888,  A  bought  an  engine  and  mill  for  $5250,  on  six  months 
credit,  or  5^  off  if  i)aid  within  90  days,  or  7^^  off  if  paid  within  30  days.  What 
amount  was  required  for  full  settlement  Nov.  7,  1888  ? 

29.  In  settling  an  estate,  an  executor  found  7^r^  of  it  to  be  invested  in 
telegraph  stock,  15f^  in  railroad  stock,  37|^  in  city  bonds,  $16750  in  real  estate, 
and  $7350  cash  in  bank.     Find  the  total  value  of  the  estate. 


172  EXAMPLES    FOR    PRACTICE   IN    PERCENTAGE. 

50.  A  farm  is  composed  of  20^  more  grazing  than  grain  land,  and  the  timber 
is  one-half  of  the  area.  How  many  acres  of  each,  if,  after  deducting  12  acres 
for  lawn  and  garden,  there  is  left  of  tlie  farm  18G0  acres  ? 

51.  A  has  20*^  less  money  than  B,  and  B  has  25't  more  than  C.  How  much 
has  C,  if  A  has  1192  ? 

32.  A  and  B  were  heirs  of  an  estate  of  #120000,  A  receiving  1  jf^  of  the  whole 
more  than  B.  For  four  years  thereafter  the  property  of  eacli  increased  at  an 
average  rate  of  9j^  per  annum.     How  much  had  each  at  the  end  of  that  time  ? 

33.  A  man  owning  62^,^^  of  a  factory,  sold  7^f^  of  his  share  for  $1050.  At 
that  rate,  what  was  the  value  of  the  factory  ? 

<?4'  What  is  the  per  cent,  of  difference  between  16f  ^  of  ^  of  a  number,  and 
25^  of  ^  of  the  same  number  ? 

35.  From  a  cheese  factory,  33630  boxes  of  cheese  were  sold  in  four  yearsj 
the  sales  of  the  second  year  having  been  30,^^  greater  than  those  for  the  first  year, 
those  of  the  third  30fc  less  than  those  of  the  second,  and  those  of  the  fourth  40^ 
greater  than  those  of  the  third.     What  were  the  sales  of  each  year  ? 

36.  In  preparing  a  prize  mixture  for  seeding  pastures,  Sibley  &  Co.  mixed 
equal  parts  of  clover  seed  and  timothy  with  33^^  as  much  orchard  grass  as  clover, 
and  33^^  as  much  red  top  as  orchard  grass.  How  many  pounds  of  each,  in  a 
consignment  of  1100  pounds  of  the  mixture  ? 

37.  Three  railroad  companies  carry  six  carloads  of  freight,  each  weighing 
20  T.  6  cwt  ,  a  distance  of  -150  miles;  the  distance  over  the  first  line  was  100 
miles,  and  that  over  the  second  125  miles.  If  the  total  charge  was  15^  per  100 
pounds,  liow  much  money  should  each  company  be  i)aid  ? 

38.  A  young  man  who  received  $21000  from  his  father,  had,  at  the  end  of  five 
years,  only  $3500  left.  "What  average  per  cent,  of  liis  inheritance  did  he  lose 
yearly  ? 

39.  My  grocery  sales  increased  20^  the  second  year,  25<^  more  the  third  year, 
and  40f^  still  more  the  fourth  year;  during  which  four  years  I  sold  $131250 
worth  of  goods.     What  was  the  amount  of  my  sales  the  first  year  ? 

40.  A  father  located  his  son  upon  a  farm,  expending  for  the  farm,  stock, 
utensils,  and  household  furniture,  $19512.50;  the  stock  cost  twice  as  much  as  the 
household  furniture,  which  cost  75^  more  than  the  farm  utensils,  and  tlie  cost 
of  the  farm  was  140^  of  the  cost  of  the  stock.     How  much  was  invested  in  each? 

Jfl.  The  general  freight  agent  of  a  railroad,  when  questioned  as  to  the  amount 
of  freight  carried  by  his  line,  replied  :  "For  the  past  four  years  our  yearly 
increase  over  previous  business  has  been  25*^;  should  this  be  shown  for  the  com- 
ing four  years,  the  amount  of  freiglit  then  carried  will  be  22070  T.  025  lb.  more 
than  double  the  amount  carried  this  year."  What  was  the  number  of  tons 
carried  four  years  ago  ? 

J^2.  A  last  will  and  testament  provided  that  three-eighths  of  the  estate 
distributed  should  go  to  the  widow,  and  the  remainder  be  so  divided  among  two 
sons  and  a  daughter  that  the  elder  son  should  receive  10;;^  more  than  the  younger, 
who  should  receive  25^  more  than  the  daughter.  What  amount  was  received  by 
each,  the  estate  being  valued  at  $58000? 


PROFIT   AND    LOSS.  173 


PROFIT    AND    LOSS. 

549.  Profit  and  Loss  treats  of  gains  or  losses  in  business  transactions. 

550.  If,  after  deducting  all  expenses  of  sale,  the  net  price  is  greater  than  the 
cost,  the  excess  is  a  Proiit  or  Gahi. 

651.  If  the  net  price  received  from  the  sale  is  less  than  full  cost,  the  differ- 
ence is  a  Loss. 

552.  The  Gross  or  Full  Cost  of  an  article  is  its  first  cost,  increased  by  all 
outlays  incident  to  its  purchase  and  holding  to  date  of  sale. 

553.  The  Net  Selling  Price  is  the  gross  selling  price,  less  all  charges  inci- 
dent to  its  sale. 

554.  In  ascertaining  profit  or  loss,  operations  are  usually  performed  by  the 
rules  of  Percentage  heretofore  explained;  but  when  the  rate  is  a  simple,  common, 
fractional  part  of  100,  it  is  more  convenient  to  use  the  equivalent  fraction  than 
the  decimal  per  cent. 

555.  Comparing  the  elements  of  Profit  and  Loss  with  those  of  Percentage, 
the  Cost  corresponds  to  the  Base;  the  Per  Cent,  of  Gam  or  Loss  to  the  Rate; 
the  whole  Gain  or  Loss  to  the  Percentage;  the  Selling  Price,  if  at  a  gain,  to 
the  Amount;  the  Selling  Price,  if  at  a  loss,  to  the  Difference. 

Remarks  —For  table  of  Aliquot  Parts,  convenient  for  use  as  common  fractional  equiva- 
lents, refer  to  page  89. 

556.  To  find  the  Profit  or  Loss,  the  Cost  and  Rate  being  given. 

Example. — An  agent  paid  195  for  a  reaper,  and  sold  it  at  a  profit  of  18^. 
What  Avas  his  gain  ? 
Operation. 

195  =  cost.  Explanation.— Since  the  agent  gained  18  per  cent,  or  18  cents  on 

.18  =  <]<,  of  gain.    1  dollar,  on  the  $95  of  cost  he  would  gain  95  times  |.18,  or  $17.10. 

$17.10  =  gain. 

VivAe.— Multiply  the  Cost  hy  the  Rate. 

Formula. — Profit  or  Loss  =  Cost  x  Kate. 

EXAMPL,ES  FOK   MKNTAL  PKACTICK. 

667.  1.  A  set  of  furniture,  costing  $60,  was  sold  at  15^'  profit.  How  mucli 
was  gained? 

2.     If  I  pay  $400  for  a  piano,  and  gain  12^  by  its  sale,  how  much  is  my  profit? 

S.  Having  paid  $7500  for  a  house,  I  sell  it  at  10^  advance  on  cost,  How 
much  do  I  gain? 


174  EXAMPLES   IN    PROFIT   AND    LOSS. 

4-  After  using  a  carriage  which  cost  me  $250,  I  was  obliged  to  sell  it  fur  20^ 
less  than  it  cost.     What  was  my  loss? 

5.  How  much  loss  do  I  sustain  by  selling  a  $200  watch  at  16^  less  than  cost? 

6.  After  paying  $1200  for  a  lot,  I  built  thereon  a  house  costing  $2800,  and 
by  selling  both  lost  S't  of  my  investment.     How  many  dollars  did  I  lose? 

7.  One  of  a  road  team  cost  $400  and  the  other  IsoOO.  How  much  is  lost,  if 
the  team  is  sold  at  25^  below  cost? 

8.  I  invested  $10500  in  Southern  lands.  If  20-^  of  the  land  proved  to  have  a 
worthless  title,  how  many  dollars  were  lost? 

9.  Since  paying  $14000  for  a  stock  of  teas,  the  price  has  advanced  5jt.  How 
much  has  the  stock  increased  in  value  ? 

KXAMPLKS   FOK   WKITTKX   PRACTICE. 

558.  -/.  Three  houses,  bought  for  $5000,  $6500,  and  $8250  respectively, 
were  sold  so  that  a  gain  of  12^  was  realized  on  the  first,  and  7^^  on  the  second, 
while  the  third  was  sold  at  6f^  below  cost.     Find  the  net  gain  or  loss? 

2..  A  stock  of  goods  costing  $15600  was  sold  at  a  loss  of  12^-^,  and  IS'v  of  the 
selling  price  was  in  bad  debts.     What  was  the  total  loss  sustained? 

3.  A  canal  boat,  loaded  with  8400  bushels  of  wheat,  collided  with  a  bridge 
pier  and  sprung  a  leak,  by  which  21^^  of  the  cargo  sustained  a  damage  equal  to 
^  of  its  value.  What  was  the  loss  sustained,  the  wheat  having  been  invoiced  at 
75^  per  bushel? 

j^  A  peddler  paid  $46.50  for  butter,  $17. 60  for  eggs,  and  $36  for  dried  berries. 
He  sold  the  butter  at  a  profit  of  16^^,  the  eggs  at  a  profit  of  20j^,  and  lost  5^ 
on  the  berries.     What  was  his  net  gain? 

5.  Having  paid  $1040  for  a  box  of  furs,  and  $18.50  expressage  on  the  same, 
I  sold  25;^  of  the  stock  at  a  gain  of  35^,  15,*?^  at  a  gain  of  20f^,  30<  at  a  loss  of 
2^,  and  the  remainder  at  cost.     How  much  did  I  gain  or  lose? 

6.  An  agent  bought  three  reapers,  paying  respectively  $90,  $120,  and  $150. 
He  sold  the  first  at  10^^  loss,  the  second  at  cost,  and  the  third  at  10^  gain.  What 
was  his  profit  by  the  transaction? 

7.  A  contractor  bought  52  M  bricks  at  $5.60  per  M,  and  sold  f  of  them  for 
f  of  their  cost,  and  for  tlie  remainder  received  $150.     What  amount  did  he  lose  ? 

8.  A  grocer  bought  7  barrels  of  sugar,  each  weighing  315  pounds,  at  61^  per 
pound,  and  sold  it  so  as  to  gain  16§^.     Find  the  amount  of  his  gain. 

9.  How  much  is  gained  by  purchasing  3  carloads  of  corn,  of  750  bushels 
each,  at  62^^  per  bushel,  and  selling  40f(p  of  it  at  a  gain  of  124^,  and  the 
remainder  at  a  gain  of  74^? 

559.  To  find  the  Cost,  the  Gain  or  Loss  and  the  Rate  of  Gain  or  Loss 
being  given. 

Example. — An  agent  gained  $17.10  by  selling  a  reaper  at  18'i  jirofit.  What 
must  he  have  paid  for  it  ? 

Operation.  Since  the  agent's  whole  gain  was  $17.10  and  since  his  i 

Rate.  Gain.  gain  on  1  dollar  of  cost  was  18r<  or  18  cents,  the  cost  must 

IS'i  =  .18  )  17.10  have  been  as  many  times  1  dollar  as  $  18  is  contained  times 

$95"  =  cost.  ^  *1  '•  10.  or  $95. 


EXAMPLES   IN    PROFIT   AND    LOSS.  ]  ?5 

^ule.— Divide  the  gain  or  loss  hij  the  -jmr  cent,  of  gain  or  loss. 
Formula. — Cost  =  Gain  or  Loss  -f-  Rate. 

EXAMPLES   FOR  MENTAL   PRACTICE. 

560.  1.     "What  was  the  cost,  if  I  lost  $15  by  selling  a  machine  \bi  below  cost? 
£?.     By  selling  a  farm  at  a  gain  of  10^,  I  realized  a  profit  of  $350.     Find  the 

cost  of  the  farm. 

S.  A  yacht  was  gold  for  11250  less  than  cost,  its  owner  thereby  losing  Vi\<f(, 
of  the  cost.     What  was  the  price  i)aid? 

Jf.  By  selling  a  consignment  of  silk  for  11^  above  the  invoice  i)rice,  a  gain  of 
$484  was  realized.     Find  the  invoice  price. 

5.  A  suit  of  clothes,  becoming  damaged,  was  sold  at  a  loss  of  13^^,  wherebv  the 
tailor  lost  $5.20.     How  much  did  the  suit  cost  when  made? 

6.  What  must  have  been  the  cost  of  a  necklace,  if  its  owner,  by  selling  it  at 
a  loss  of  15^^,  received  $45  less  than  it  cost? 

7.  By  selling  a  coach  for  $G3  above  cost,  I  gained  1%  on  my  })urchase  price. 
How  much  did  it  cost? 

S.  Having  received  $105  more  for  a  house  than  its  cost,  I  find  my  jirofit  to 
be  lOj^      How  much  did  I  pay  for  the  house? 

9.  A  oook-seller  lost  65^-  on  an  album,  and  thereby  sustained  a  loss  of  65^. 
Find  the  cost. 

EXAMPLES   FOR  WRITTEN   PRACTICE. 

561.  1.  What  must  have  been  the  cost  of  a  watch  and  chain,  if  $6.90  was 
lost  by  selling  them  at  12^  below  cost? 

.?.  A  dealer  sold  a  piano  at  25ftf  profit,  and  witli  the  jiroceeds  bought  another 
which  he  sold  at  20^^^  profit,  realizing  a  total  gain  of  $250.  What  was  the  cost 
of  each? 

3.  By  selling  a  lot  for  $1680,  I  received  40j^  more  tlian  twice  its  cost.  At 
what  price  did  1  purchase  it? 

Jf.  A  sells  a  horse  to  B  and  gains  15,<.  If  13  i)aid  25ffc  of  $420  more  for  the 
horse  than  A  did,  at  what  price  did  A  buy  it? 

5.  Having  bouglit  a  house  of  A  at  12^,^  less  than  it  cost  liim,  I  added  $4;)0  in 
repairs,  and  sold  it  for  $7293,  thereby  gaining  10<^  on  my  investment.  How 
mucli  did  the  house  cost  A? 

6.  A  miller's  gain  in  business  for  four  years  aggregates  ^l^f^  of  his  capital.  If 
his  gain  is  $3000,  and  he  withdraws  his  gain  and  capital  and  invests  it  in  a  farm, 
at  $55  per  acre,  how  many  acres  can  he  buy? 

7.  A  merchant  bought  goods  and  paid  freight  on  them  equal  to  12,''.'  of  their 
first  cost;  he  then  sold  them  at  m  profit  on  the  full  cost,  receiving  QO'^  of  the 
price  in  cash  and  a  note  for  $1309,  the  amount  unpaid.  Wluit  was  the  first 
cost  of  the  goods? 

8.  A  peddler  sold  25^  of  a  purchase  of  butter  at  16j^  profit,  and  the  remanider 
at  165^  profit.     What  was  the  cost,  if  the  total  gain  was  $39.60? 

9.  A  dealer  sold  35^  of  a  purchase  of  leather  at  141'^,'  jirofit,  and  the  remainder 
at  0%  loss.     If  his  net  gain  was  $87.50,  what  must  have  been  the  cost? 


170  EXAMPLES   IN   PROFIT   AXD   LOSS. 

562.  To  find  the  Rate  of  Profit  or  Loss,  the  Cost  and  the  Profit  or  Loss  being 
given. 

Example. — An  agent  gained  $17.10  by  selling  a  reaper  which  cost  him  $95. 
What  was  his  jier  cent,  of  gain  ? 

Operation. 
Cost.      Gain. 

$95  )  17.  K>  (  .18  =  18^  Explanation.— If  95  dollars  of  cost  gain  $17.10,  1  dollar  of 

95  cost  would  gain  as  much  as  95  is  contained  times  in  17.10,  or  .18, 

^Q  equal  to  18  per  cent. 

760 

Unle.— Divide  the  profit  or  loss  by  the  cost. 

Formula. — Per  Cent,  of  Profit  or  Loss  =  Profit  or  Loss  -^  Cost. 

EXAMPI.ES  FOR  MENTAL   PRACTICE. 

563.  -?.     I  gained  $12.50  on  what  cost  me  $125.     Find  my  rate  per  cent,  of 
gain. 

2.     I  bought  a  bicycle  for  $150,  and  sold  it  for  $7.50  below  cost.     What  per 
cent,  did  I  lose? 

S.     What  per  cent,  is  lost  by  selling  a  $5  book  at  62-|^  below  its  cost? 

4.  A  safe  costing  $380  was  sold  at  a  loss  of  $76.     Find  the  loss  per  cent. 

5.  A  gold  pen  cost  $2,  and  after  being  tested  and  found  imperfect,  was  sold  as 
old  gold  for  $1.     Find  the  per  cent,  of  loss. 

6.  What  per  cent,  of  gain  is  realized  by  buying  a  horse  for  $300,  and  selling 
it  at  an  advance  of  $100? 

7.  Find  the  per  cent,  of  gain, on  a  section  of  Dakota  prairie,  bought  at  $4  per 
acre,  and  sold  at  $10  per  acre. 

8.  An  Ohio  river  steamer  costing  $100000  was  sold  for  $9500  profit.     Find 
the  per  cent,  of  profit. 

9.  A  Vermont  manufacturer,  having  invested  $40000,  gained  $8250  each 
year.     What  was  his  per  cent,  of  gain  per  annum? 

EXAMPLES  FOR  WRITTEN   PRACTICE. 

564.  1.     What  per  cent,  is  gained  by  selling  an  article  for  2-k  times  its  cost? 

2.  I  bought  a  quantity  of  cloth  at  $1.60  per  yard,  and  sold  it  at  $2  per  yard. 
What  was  my  per  cent,  of  gain? 

3.  A  speculator  bought  wheat  at  80^-  per  bushel,  and  oats  at  32{i^-  per  bushel. 
If  he  sold  the  wheat  at  90^'  per  bushel,  and  the  oats  at  40^  per  bushel,  on  which 
would  he  make  the  greater  per  cent.,  and  how  much? 

4.  If  a  boy  sells  three  apples  for  what  four  cost  him,  what  per  cent,  does  he 
gain? 

6.  Four-fifths  of  a  stock  was  sold  at  45^  loss,  and  the  remainder  at  225^ 
profit.     What  was  the  per  cent,  of  net  loss  or  net  gain  on  the  stock? 

6'.  Paper  bought  at  $2.70  per  ream,  and  retailed  at  1^  per  sheet,  will  yield 
what  per  cent,  of  profit? 


EXAMPLES    IN   PROFIT   AND    LOSS.  177 

7.  Potatoes  costing  $1.35  per  barrel,  and  sold  at  11.62  per  barrel,  will  net 
what  per  cent,  of  gain  ? 

8.  A  wood  dealer,  after  buying  8  car  loads  of  mixed  wood,  of  IG  cords  each, 
at  $5  per  cord,  sorted  it  and  sold  35^  of  it  at  1%\%  gain,  35^  of  it  at  10^  gain,  and 
the  remainder  at  20f«  gain?     What  was  his  average  per  cent,  of  gain? 

9.  If  33^^  of  a  barrel  of  salt  be  sold  at  33^^  profit,  and  the  remainder  be  sold 
at  cost,  what  per  cent,  of  profit  is  realized  on  the  whole? 

10.  An  agent  sold  a  sewing  machine  for  $45.70,  and  thereby  gained  $18.28. 
What  per  cent,  did  he  gain? 

11.  If  \  of  an  article  is  sold  for  what  f  of  it  cost,  what  is  the  loss  per  cent.? 

12.  If  I  sell  ^  of  an  article  for  what  ^  of  it  cost,  what  is  my  rate  of  gain? 

13.  A  drover,  buying  125  beeves  at  the  rate  of  $55  per  head,  and  78  at  $G2.50 
per  head,  sold  the  lot  at  a  profit  of  $2115.     What  was  his  per  cent,  of  gain? 

U.  A  cargo  of  lumber  cost  $3000.  If  ^  of  it  is  sold  for  82000,  4  of  the 
remainder  for  $1250,  and  what  is  left  for  $420,  what  is  the  per  cent,  of  gain  or 
loss  by  the  transaction? 

16.  Oil  bought  at  81^^"  per  barrel  is  sold  at  86|^'.  If  ^^'  per  barrel  is  allowed 
for  expenses,  what  must  have  been  the  investment,  the  gain  having  been  $1350? 

565.  To  find  the  Cost,  the  Selling  Price  and  the  Rate  per  cent,  of  Profit  or 
Loss  being  given. 

Rules.— i.    Divide  the  selling  price  by  1  plus  the  rate  of  gain.    Or, 
2.    Divide  the  selling  price  by  1  minus  the  rate  of  loss. 

Formulas  —  •!  ^'  ^^^^  ~  ^^^^^^S  ^^^ce  -^  1  +  Per  Cent,  of  Gain. 
(  b.  Cost  =  Selling  Price  ~  1  —  Per  Cent,  of  Loss. 
« 

EXAMPLES   FOR  MENTAL   PRACTICE. 

566.  1.     A  buggy  was  sold  for  $105,  at  a  gain  of  5^.     What  the  the  cost? 

2.  What  must  have  been  the  cost  of  a  harness  sold  at  40^  loss,  if  $24  were 
received  for  it? 

3.  Find  the  cost  of  making  a  suit  of  clothes,  if  20^  is  gained  by  selling  it 
at  $18. 

Jf.     Find  the  cost  of  a  watch  that  sold  at  a  profit  of  16S^  and  brought  $87.50. 

6.  I  sold  a  house  for  125^  profit,  receiving  therefor  $2250.  Wliat  was  the 
price  paid? 

6.  If  $15360  is  realized  on  a  stock  of  goods  after  it  has  been  damaged  40^, 
what  was  its  value  before  being  damaged? 

EXAMPLES   FOR  WRITTEN   PRACTICE. 

567.  1.  One  of  a  pair  of  horses  was  sold  for  $180,  at  a  loss  of  12A^;  the  other 
was  sold  for  $200,  at  a  gain  of  25^.     What  did  the  pair  cost? 

2.  A  fruit  dealer,  after  losing  16!  of  his  apples  by  frost,  has  147^  barrels  left. 
If  he  bought  his  stock  at  $2.50  per  barrel,  what  was  his  outlay? 

3.  What  was  the  original  value  of  Calumet  copper  mining  stock,  which,  when 
«old  at  a  gain  of  175^,  brought  $20625? 

12 


178  REVIEW    OF   THE    PRINCIPLES   OF    PROFIT   AXD   LOSS. 

Jf..  A  paid  6^  tax  on  his  income.  What  was  his  income,  if,  after  paying  the 
tax,  the  remainder  equalled  $7050.94. 

5.  A  dairy  produced  20^  more  cheese  in  March  than  in  February,  "What 
was  the  jiroduct  for  March,  if  that  for  the  two  months  was  1980  pounds  ? 

6.  I  sold  a  house  to  A  at  a  profit  of  lO't;  he  sold  it  to  B.  gaining  \h^\  and  B, 
by  selling  it  to  C  for  $6072,  gained  20^  on  his  purchase.  How  much  did  the 
house  cost  me? 


REVIEW    OF    THE    PRINCIPLES    OF    PROFIT    AND    LOSS. 

568.  -?.  To  find  the  gain,  the  cost  and  per  cent,  of  gain  being  given. 
Rule. — Multiply  the  cost  by  the  j^er  cent,  of  gain. 

2.  To  find  the  loss,  the  cost  and  percent,  of  loss  being  given.  Rule. — Mul- 
tiply the  cost  by  the  per  cent,  of  loss. 

3.  To  find  the  selling  price,  the  cost  and  gain  being  given.  Rule. — Add 
the  gain  to  the  cost. 

4.  To  find  the  selling  price,  tlie  cost  and  loss  being  given.  Rule. — Sub- 
tract the  loss  from  the  cost. 

5.  To  find  the  cost,  the  gain  and  per  cent,  of  gain  being  given.  Rule. — 
Divide  the  gain  by  the  per  cent,  of  gain. 

6.  To  find  the  cost,  the  lossand  per  cent,  of  loss  being  given.  Rule. — Divide 
the  loss  by  the  per  cent,  of  loss. 

7.  To  find  the  selling  price,  the  gain  and  per  cent,  of  gain  being  given. 
Rule. — Divide  the  gain  by  the  ])er  cent,  of  gain,  and  to  the  quotient  add  the  gain. 

8.  To  find  tlie  selling  price,  the  loss  and  per  cent,  of  loss  being  given. 
Rule. — Divide  the  loss  by  the  per  cent,  of  loss,  and  froin  the  quotient  subtract 
the  loss. 

9.  To  find  the  per  cent,  of  gain,  the  gain  and  cost  being  given.  Rule. — 
Divide  the  gain  by  the  cost. 

10.  To  find  the  per  cent,  of  loss,  the  loss  and  cost  being  given.  Rule. — Divide 
the  loss  by  the  cost. 

11.  To  find  the  per  cent,  of  gain,  the  selling  price  and  gain  being  given. 
Rule. — Subtract  the  gain  from  the  selling  price  and  divide  the  gain  hu  the 
quotient. 

12.  To  find  the  per  cent,  of  loss,  the  selling  jirice  and  loss  being  given. 
Rule. — Add  the  loss  to  the  selling  price,  and  divide  the  loss  by  the  sum  obtained. 

MISCELLANEOUS  EXAitPLES. 

569.  1.    What  is  that  sum  of  money  of  which  50^  is  $19.20  more  than  37^j^? 

2.  What  amount  of  money  must  an  attorney  collect,  in  order  that  he  may 
pay  over  to  his  principal  $475,  and  retain  b<^  for  his  services? 

3.  A  woman  is  72  years  old,  and  16^|<^  of  her  age  is  25,^  of  the  age  of  her 
daughter.     Find  the  daughter's  age. 

4.  Gunpowder  is  made  of  f  nitre,  and  the  remainder  of  equnl  parts  of 
sulphur  and  charcoal.     Find  the  per  cent,  of  each. 


MISCELLANEOUS    EXAMPLES    IN    PROFIT   AND   LOSS.  179 

5.  A  milkman  increased  his  herd  of  cows  by  a  purchase  of  36,  which  was  45^ 
of  the  whole  number  he  then  owned.     How  many  had  he  before  buying  the  last  lot? 

6.  If  I  make  a  profit  of  16f^  by  selling  a  horse  at  87.50  above  cost,  how 
much  must  I  have  advanced  on  the  cost  to  have  realized  a  profit  of  25f^? 

7.  Two  persons  contributed  $2100  towards  a  business  venture,  from  which 
their  part  of  the  gain  was  $350.  If  of  this  gain  the  share  of  one  was  $70  more 
than  that  of  the  other,  what  part  of  the  original  contribution  must  have  been 
made  by  each  ? 

8.  How  much  money  must  be  invested  in  notes,  at  4|«^  below  their  face  value, 
in  order  that,  when  sold  at  ?>io  above  their  face,  a  profit  of  $225  may  be  realized  ? 

9.  I  bought  a  warehouse  of  Brown  for  12^^  less  than  it  cost  him,  and  sold  it 
for  16f^  more  than  it  cost  him,  gaining  thereby  $963. GO.  How  much  did  I  pay 
for  the  warehouse  ? 

10.  What  per  cent,  is  gained  by  buying  pork  at  $17.50  per  barrel,  and  retail- 
ing it  at  12^  per  pound  ? 

11.  A  lady  wishing  to  sell  her  piano,  asked  15^  more  than  it  cost,  but  finally 
sold  it  at  12.T^  less  than  her  asking  price.  What  did  the  piano  cost,  if  by  its 
sale  she  gained  155  ? 

12.  Having  bought  75  barrels  of  ajiples  for  $187.50,  I  sold  them  at  a  loss  of 
20^.     How  much  did  I  receive  per  barrel  ? 

13.  A  sells  a  steam  tug  to  B,  gaining  12^^^,  and  B  sells  it  to  C  for  $4130,  and 
makes  a  profit  of  18^.     How  much  did  the  tug  cost  A  ? 

III..  What  per  cent,  is  lost  on  an  article  that  is  sold  for  two-thirds  of  its 
cost  ? 

15.  A  farmer,  after  selling  1760  barrels  of  apples,  had  20^  of  his  crop  left. 
How  many  barrels  had  he  at  first? 

16.  I  lost  25^  of  a  consignment  of  berries.  At  what  per  cent,  of  profit  must 
the  remainder  be  sold,  in  order  that  I  may  gain  10^  on  the  whole  ? 

n.  A  Texas  farm  of  160  acres  was  bought  at  $15  jier  acre;  8354  were  paid  for 
fencing,  $480  for  breaking,  $626  for  a  house,  and  $220  for  a  barn.  At  what 
price  per  acre  must  it  be  sold,  to  realize  a  net  profit  of  25,'^  on  the  investment  ? 

18.  King  sold  his  wheel  at  33^^  gain,  and  with  the  money  bought  another, 
which  he  sold  at  a  loss  of  25^,  receiving  therefor  $120.  Did  he  gain  or  lose, 
and  how  much  ? 

19.  What  per  cent,  more  is  \  than  f  ? 

20.  Cloth,  bought  at  $4  per  yard,  must  be  marked  at  what  price  in  order  that 
the  seller  may  make  a  reduction  of  10^  from  the  asking  price  and  still  gain  124^ 
on  the  cost  ? 

21.  If  25^  of  the  selling  price  is  gain,  what  is. the  per  cent,  of  gain  ? 

22.  I  sell  f  of  a  stock  of  goods  for  $27,  thereby  losing  20^.  For  what  must 
I  sell  the  remainder,  to  make  a  profit  of  20^  on  the  whole  ? 

23.  If  30c^  of  a  farm  sold  at  33^^  gain,  and  30f^  of  the  remainder  at  15^  gain, 
how  much  was  the  total  gain,  if  the  remainder  was  sold  at  cost  for  $7350  ? 

^^.  What  per  cent,  of  cost  is  realized  on  goods  marked  25^^  advance  and  sold 
at  20^  off  from  the  marked  price  ? 


180  MISCELLANEOUS    EXAMPLES   IN"   PROFIT   AND    LOSS. 

25.  A  biinker  bought  a  mortgage  at  7^^  less  than  its  face  value,  and  sold  it 
for  Z'/t  more  than  its  face  value,  thereby  gaining  $981.75.  What  was  the  face 
value  of  the  mortgage  ? 

26.  At  what  jn-ice  should  damaged  goods  be  marked  to  lose  25^,  the  first 
cost  having  been  36^  per  yard  ? 

27.  A  man  sold  a  carriage  and  gained  25^,  and  with  the  proceeds  bought 
iinother,  wliicli  he  sold  at  a  profit  of  lO,'^,  thus  realizing  a  total  gain  of  $75. 
"What  did  he  pay  for  eacli  ? 

28.  If  I  sell  f  of  an  acre  of  land  for  what  %  of  it  cost,  what  Avill  be  my  gain 
or  loss  per  cent.  ? 

29.  21-^^  was  lost  by  selling  an  engine  for  $2355.  How  much  would  it  have 
brought  had  it  been  sold  at  a  loss  of  XOfc  ? 

30.  "What  price  must  be  asked  for  1000  pounds  of  coffee,  costing  18^  per 
pound,  in  order  tliat  tlio  seller  may  deduct  lO,'^  from  the  asking  price  for  bad 
debts,  allow  1G|^  for  loss  in  roasting,  and  still  gain  20,^^  on  the  cost? 

31.  B  and  C  each  invested  an  equal  amount  of  money  in  business;  B  gained 
12-i^  on  Ills  investment,  and  C  lost  $5275  ;  C's  money  was  tlien  42^  of  B's. 
How  many  dollars  did  eacli  invest  ? 

32.  A  trader  lost  33^f^  on  20^  of  an  investment,  and  gained  12^^  on  the 
remainder,  thus  realizing  a  net  gain  of  $1000.  Had  he  gained  20^  on  -J,  and 
lost  2")'^  on  the  remainder,  what  would  have  been  his  net  profit  ? 

33.  A  manufacturing  company's  per  cent,  of  gain  on  a  self-binder  was  25^  less 
than  that  of  tlie  general  agent;  the  general  agent's  profit  was  20^,  he  thereby 
gaining  $25.30.     "What  did  it  cost  to  make  the  machine  ? 

31)..  Of  a  cargo  of  8000  bushels  of  oats,  costing  35^-  per  busliel,  25j^  was 
destroyed  by  fire.  What  per  cent,  will  be  gained  or  lost,  if  the  remainder  of  the 
oats  are  sold  at  45^  per  busliel  ? 

35  For  Avhat  must  hay  be  sold  per  ton,  to  gain  16|^  if,  by  selling  it  at  $18 
per  ton.  there  is  a  gain  of  25^  ? 

30.  Jones  sold  \  of  a  stock  of  goods  at  cost,  \  at  a  gain  of  35^,  ^  at  a  loss  of 
25^,  and  -^^  at  a  gain  of  10^.  At  what  per  cent,  of  its  cost  must  he  sell  the 
remainder  to  net  cost  on  the  whole  ? 

37.  After  a  carriage  had  been  used  two  years,  it  was  sold  for  $5  less  than  it 
cost,  the  seller  thereby  sustaining  a  loss  of  'd^'fo  of  tlie  selling  price.  How  much 
was  the  first  cost  of  the  carriage  ? 

38.  li  oranges  cost  $1.80  per  liundred,  at  what  price  must  tliey  be  marked 
to  ensure  a  gain  of  20^,  and  make  allowance  for  28^  decay,  and  25,^  bad  debts  in 
selhng  ? 

39.  Having  paid  40^-  per  pound  for  tea,  at  Avhat  retail  price  must  it  be  marked, 
that  I  may  allow  12^^  for  bad  debts  and  gain  40^  on  the  cost  ? 

Jf.0.  Six  wheel-rakes  were  sold  for  $21  each;  three  of  them  at  a  gain  of  20j^, 
and  tlie  others  at  a  loss  of  20je.     "What  was  the  net  gain  or  loss  ? 

Jf.1.  A  stock  of  goods  is  marked  22^^  advance  on  cost,  but  becoming  damaged, 
is  sold  at  20^  discount  on  the  marked  price,  whereby  a  loss  of  $1180.40  is  sus- 
tained.    "What  was  the  cost  of  tlu;  goods  ? 


MISCELfcANEOUS   EXAMPLES   IN    PROFIT   AXD    LOSS.  181 

Ji2.  My  retail  price  of  Axminster  carpet  is  $3.50  per  yard,  by  which  I  gain 
25^.  If  I  sell  at  wholesale,  at  a  discount  of  %o^o  from  the  retail  price,  how  much 
do  I  receive  per  yard.  "What  is  my  per  cent,  of  gain  or  loss,  and  how  much  is 
my  actual  gain  or  loss  by  selling  1000  yards  at  wholesale  ? 

JfS.  If  the  loss  equalled  \  of  the  selling  price,  what  was  the  per  cent,  of 
loss  ? 

Ji4.  A  grocer  bought  200  quarts  of  berries,  at  11^^  per  quart,  and  150  quarts 
of  cherries,  at  6^^  per  quart.  Having  sold  the  cherries  at  a  loss  of  30fc,  for  how 
much  per  quart  must  he  sell  the  berries,  to  gain  15j^  on  the  whole  ? 

Jf.G.  A  sells  two  horses  to  B  at  an  advance  of  IGf;?;,  B  sells  them  to  C  at  an 
advance  of  25,'?^,  and  C  sells  them  to  D  for  $735,  thereby  making  a  profit  of  20^. 
IIow  much  did  A  pay  for  the  horses  ? 

Jfi.     Having  bought  48  pounds  of  coffee,  at  the  rate  of  34-  i:>ounds  for  91^,  and 
84  pounds  more  at  the  rate  of  7  pouads  for  $1.26,  I  sold  the  lot  at  the  rate  of 
9  pounds  for  $1.53.     What  was  my  per  cent,  of  gain  or  loss  ? 
'  Jfl.     By  selling  at  a  loss  of  6^  per  yard  I  get  87+^^  of  the  cost  of  cloth.     What 
per  cent,  of  the  cost  would  I  have  received  had  I  lost  %<P  per  yard  ? 

Jf8.  If  15^  is  lost  by  selling  suits  at  $17  each,  how  much  would  be  gained  by 
selling  them  at  15^  profit  ? 

Jt.9.  The  price  of  a  suit  of  clothes  having  been  marked  down  20j^  or  to  $27, 
the  dealer,  in  order  to  effect  a  sale,  discounted  again  15^^^,  and  still  by  the  sale 
made  a  profit  of  14f  ^.      What  per  cent,  above  cost  was  the  suit  originally  marked  ? 

50.  An  Iowa  farm  passed  through  the  hands  of  five  owners,  each  of  whom 
in  succession  gained  20^':^  by  its  purchase  and  sale.  If  the  average  gain  was 
$1488.32,  what  was  its  first  cost,  and  what  was  its  final  selling  price  ? 

51.  By  selling  a  stock  of  goods  at  20^'i^  below  cost,  I  received  $150  less  tluin  I 
would  have  received  had  I  sold  tlie  goods  at  20^  above  cost.  Wluit  should  the 
goods  have  sold  for  to  gain  20,ro  ? 

52.  The  first  cost  of  Parisian  goods  purchased  through  an  agent  was  increased 
18,<^  by  the  charges  of  the  agent,  the  freight,  and  the  import  duties;  I  sold  the 
goods  at  25,^  advance  on  full  cost,  thereby  gaining  $1785.     Find  the  first  cost. 

5S.  After  iising  a  carriage  for  two  years,  I  sold  it  for  3^^  of  its  selling  price 
less  than  it  cost,  thereby  losing  $5.  How  much  would  it  have  brought,  had  the 
amount  received  for  it  been  "i^^o  of  the  selling  price  more  than  it  cost  ? 

5Jt.  An  agent  bought  a  reaper  at  20^  off  from  the  wholesale  price,  and  sold 
it  at  an  advance  of  30^,  thereby  gaining  $37.50.  If  the  wholesale  price  was  25^ 
above  the  cost  of  manufacture,  what  was  the  cost  to  the  manufacturer? 

55.  I  sold  a  house  at  25^  profit,  and  invested  the  proceeds  in  dry  goods,  on 
which  I  lost  12^^  ;  I  invested  the  proceeds  from  the  sales  of  dry  goods  in  stocks, 
on  which  I  lost  10<^.     What  was  my  net  gain  or  loss  per  cent.  ? 

56.  Having  paid  a  retailer  $138.60  for  a  set  of  furniture,  I  ascertain  that  by 
selling  to  me  he  gained  12^^,  that  the  wholesaler  of  whom  he  bought  gained  10^, 
that  the  jobber  by  selling  to  the  wholesaler  gained  16|^,  and  that  the  manufac- 
turer sold  to  the  jobber  at  20^  above  its  first  cost.  How  much  more  than  its 
first  cost  did  I  pay  ? 


182  MISCELLANEOUS    EXAMPLES    IX    PROFIT    AND    LOSS. 

57.  I  wish  to  line  the  carpet  of  a  room  21  ft.  long  and  18  ft.  wide  with  duck 
f  of  a  yard  in  Avidth.  How  many  yards  will  be  reqnired,  if  it  shrink  10,'^  in 
width  and  hi>  in  length  ?  If  tlie  carpet  be  laid  lengthwise  of  the  room,  and  be 
furnished  at  $2.25  per  yard,  f  of  a  yard  wide,  and  the  duck,  before  shrinking,  at 
200  per  square  yard,  and  a  i)r()fit  of  IGf*^  be  realized  on  both,  Avhat  Avill  be  the  gain? 

58.  If  I  pay  83.20  for,  20  gal.  of  vinegar,  how  many  gallons  of  water  must 
be  added,  that  40,'^  profit  may  be  realized  by  selling  it  at  \b<f!  per  gallon  ? 

59.  A  huckster  sold  a  quantity  of  potatoes  and  onions,  gaining  37^^  on  the 
onions  and  25ffc  on  the  potatoes,  33J,'*  of  his  profit  being  realized  on  the  potatoes. 
At  what  price  was  each  sold,  if  the  total  gain  was  $450  ? 

60.  What  price  each  must  be  asked  for  cocoanuts,  costing  S4  i)er  C,  that  an 
allowance  of  l<o'^'t  for  breakage,  20,^^  for  decay,  and  ll|,ff  for  bad  debts  may  be 
made,  and  still  a  profit  of  33^'^'  be  realized  ? 

61.  A  tree  agent  sold  apple  and  pear  trees  for  $2187.50;  he  gained  16|^  on 
the  apple,  and  37|,'o  on  the  pear  trees,  receiving  for  the  pear  75^  as  much  money 
as  for  the  apple  trees.     Find  the  cost  of  each  kind. 

62.  A  dry  goods  house  bought  a  stock  of  goods,  and  sold  \  of  it  at  25^  profit, 
I  of  it  at  20^  profit,  ^  of  it  at  16f/^  loss,  ^  of  it  at  12^^  gain,  and  the  remainder, 
which  cost  $4549.25,  at  15^  gain.  What  Avas  the  net  gain  or  loss,  and  the  jier. 
cent,  of  gain  or  loss,  on  the  entire  stock  ? 

63.  A  butcher  paid  equal  amounts  of  money  for  calves,  pigs,  and  sheep;  he 
cleared  14*^  on  the  calves,  10,l!  on  the  pigs,  and  lost  30^  on  the  sheep.  How 
many  dollars  Avere  i)aid  for  each  kind  of  stock,  the  total  amount  recei\'ed  having 
been  $1336.50? 

64.  I  sold  my  house  to  B  and  lost  10^  of  its  cost;  B  expended  $375  for  repairs 
and  sold  it  to  C  at  120'^  of  its  full  cost  to  him;  C  expended  $525  in  enlarging 
the  house,  and  then  sold  it  for  $6354,  thereby  making  a  profit  of  20^  of  its  full 
cost.     HoAv  much  did  I  i)ay  for  the  property  ? 

65.  A  speculator,  investing  equal  sums  in  corn  and  Avheat,  gained  $2713.50 
more  on  the  corn  than  on  the  Avheat.  If  he  gained  10|^  on  the  Avlieat  and  15^ 
on  the  corn,  hoAV  many  bushels  of  each  must  have  been  purchased,  the  corn 
having  been  bought  at  60^'  per  bushel  and  the  Avheat  at  80^  per  bushel  ? 

66.  A  drover  bought  50  horses,  cows,  and  sheep  for  $870;  the  number  of  coavs 
was  600^  of  the  number  of  horses,  and  the  number  of  sheep  was  300j^  of  the 
number  of  cows;  the  horses  cost  200$^,  and  the  sheep  20^,  as  much  as  the  cows. 
If  the  entire  purchase  was  sold  at  a  profit  of  20'?^,  hoAV  much  Avas  received  i)er 
head  for  each  kind  ? 


TRADE    DISCOUNT.  183 


TRADE    DISCOUNT. 

570.  Discount  is  the  allowance  made  for  the  i)ayment  of  a  debt  before  it 
becomes  due. 

571.  Trade  Discount  is  the  allowance  made  by  manufacturers  and  mer- 
chants upon  tlieir  fixed  or  list  prices. 

Remahks. — 1.  It  is  customary  iu  many  branches  of  business  for  merchants  and  manu- 
facturers to  have  fixed  price  lists  of  their  goods,  and  when  the  market  varies,  instead  of 
changing  the  price  list,  to  change  the  rate  of  discount. 

2.  Business  houses  usually  announce  their  terms  upon  their  "  bill-heads,"  as,  "  Terms,  3 
months,  or  5;«  off  for  cash;"  "  Terms,  60  days,  or  3;?  discount  in  10  days,"  etc.  When  bills 
are  paid  before  maturity,  legal  interest  for  the  remainder  of  the  time  is  usually  deducted. 

572.  There  may  be  more  than  one  Trade  Discount,  and  they  are  then  known 
as  a  Discount  Series. 

573.  Trade  Discount  is  computed  by  the  rules  of  percentage,  on  the  marked 
price  as  a  base.  When  a  series  of  discounts  is  allowed,  the  first  only  is  so  com- 
puted, and  in  every  subsequent  discount  the  remainder,  after  each  preceding 
discount,  is  regarded  as  tlie  base. 

574.  To  find  the  Selling  Price,  the  List  Price  and  Discount  Series  being  given. 

Example  (first  illustration). — The  list  price  of  a  sewing  machine  is  ^00. 
"Wliat  is  the  net  selling  price,  if  a  discount  of  40,^0  is  allowed  ? 
Operation. 
$  60  =  list  price. 

••^^  =  ^  of  discount.  Explanation. — Since  the  discount  is  40  per  cent.,  and 

f  24  =  discount.  *^^  ^^^^  price,  or  base,  is  .$60,  the  discount  to  be  deducted 

will  be  40  per  cent,  of  $60,  or  $24 ;  and  the  net  price  will 
$  60  =  cost.  be  $60  minus  $24,  which  equals  $36. 

24  =  discount. 
$  36  =  2ict  selling  price. 

Example  (second  illustration).— Tlie  list  price  of  a  threshing  machine  is  1900. 
TVhat  is  the  net  price,  if  a  discount  scries  of  25^,  20,<,  and  10^  is  allowed  ? 
Operation. 
$900  =  list  price. 

~^^  —  ^^^-«  or  :^  =  1st  discount.  Explanation.— From  the  list  price  take  the 

$075  =  rem.  after  1st  discount.  ^^^  discount,  and  make  each  remainder  the  base 

135  =  20'^    or  4  =  2d  discount  ^^^  ^^^^  succeeding  discount.     The  last  remainder 

rr7~                     .  "^        ^    , .  will  be  the  net  price. 
$o40  =  rem.  after  2d  discount. 

54  =  10^,  or  -fu-  =  3d  discount. 
$480  =  rem,  after  od  discount,  or  net  price. 
Remark. — Iu  like  manner  treat  any  series  of  discounts. 


184  EXAMPLES   IN   TRADE    DISCOUNT. 

Rule. — Deduct  the  first  discount  from  the  list  price,  and  each  subse- 
ifuent  discount  from  each  successive  remainder. 

CXAMPIJSS  FOK  PKACTICK. 

575.  1.  What  is  the  selling  price  per  dozen  of  hats,  listed  at  136,  and 
discounted  20f^  and  15,<^  ? 

2.  Find  the  net  price  of  a  ton  of  fence  wire,  listed  at  9'/  per  pound,  and  sold 
at  T0,<  and  hio  off. 

S.  Find  the  net  cost  to  the  purchaser  of  a  bill  of  goods  invoiced  at  81100, 
from  which  discounts  of  20f^  and  'i.h'fo  Avere  allowed. 

4.  An  invoice  of  silk  amounting  to  $12000  was  sold  Sept.  21,  1888,  at  a 
discount  of  25,^^,  20^,  and  12^,^^,  with  a  further  discount  of  I'd'lo  to  be  allowed  if 
paid  within  30  davs.     How  much  cash  will  pay  the  bill  Oct.  15,  1888  ? 

5.  Having  bought  merchandise  at  25'^  and  ISf*  discount  from  the  list  ])rice 
of  $1500,  I  sell  it  at  Ib^t,  15c?,  and  10,^  from  the  same  list  price.  Do  I  gain  or 
lose,  and  how  much  ? 

6.  A  wholesale  dealer  offers  cloth  at  §2. -40  j^er  yard,  subject  to  a  discount  of 
25^,  20,^^,  10^,  and  5^.     How  many  yards  can  be  bought  for  $246.24  ? 

7.  What  is  the  net  cost  of  a  bill  of  goods  invoiced  at  $2150,  and  sold  at  a 
discount  of  15'^,  10^,  hi,  and  Z<  ? 

8.  Three  drummers.  A,  B  and  C,  offer  me  the  same  grade  of  goods  at  the 
same  list  price.  A  offers  to  discount  25^  and  15^;  B  20^^  and  20^;  and  C  15^^ 
15^,  and  10^.  With  which  will  it  be  most  advantageous  for  me  to  deal,  and 
how  much  would  I  save  from  a  list  price  of  $200  ? 

Remarks. — 1.  It  is  often  convenient  in  finding  the  net  price  to  multiply  the  list  price  by  1 
minus  the  first  discount,  the  remainder  by  1  minus  the  next,  and  so  on. 

2.  The  order  in  which  the  discovmts  of  any  series  are  considered  is  not  material,  a  series  of 
25,  15,  and  10  being  the  same  as  one  of  15,  10,  and  25,  or  of  10,  25,  and  15. 

576.  To  find  the  Price  at  which  Goods  must  be  Marked  to  Insure  a  Given 
Per  Cent,  of  Profit  or  Loss,  the  Cost  and  Discount  Series  being  given. 

Example  (first  illustration). — Having  bought  goods  for  $105,  at  what  price 
must  they  be  marked  to  allow  a  discount  of  %h<:,  and  still  make  a  profit  of  10,^  ? 

Explanation. — The  cost,  $105,  is  100  per 
Operation  cent,  of  itself ;  the  rate  of  discount  to  be  al- 

.-„_  ,1  L  lowed  is  25  per  cent.;  100  percent,  minus  25 

'  per  cent.,  or  75  per  cent.,  is  the  per  cent,  which 

•^^^  =  ^  to  be  gamed.  ^j^^  ^^^^  ^  ^  insured  is  of  the  price  to  be 

$10.50  =  giiiii  to  be  insured.  asked.     And  if  10  per  cent,  must  be  insured, 

105  00  =  cost.  ^^^  goods  must  actually  bring  10  per  cent.,  or 

7       ,-.  •       .     i_  T  $10..")0  more  than  cost,  or  $115.50.     And  since 

$115.50  sellmg  price  to  be  insured.  ,  ,    ..       t  ^r  ,  •  .    i         ^   , 

*  o  i  a  deduction  of  25  per  cent,  is  to  be  made  from 

n,f.  X  A^  ^  -  -Q  the  iiskiug  price,  the  selling  price,  $115.50,  will 

'     '- —  _  .  be  only   75    per  cent,    of    the   asking  price. 

$154     asking  price.  Therefore,  divide  $115.50  by  .75,  and  the  quo- 

tient, $154,  will  be  the  asking  price. 


EXAMPLES   IN   TRADE   DISCOUNT.  185 

Example  (second  illustration). — A  seal  sacque  cost  a  manufacturer  $240.  At 
what  price  must  it  be  marked,  that  a  discount  series  of  25^,  20^,  and  20^  maj 
be  allowed,  and  he  still  make  a  profit  of  30^  ? 

Operation, 
$240  =  cost  or  base.  $1.00  =  ^  of  price  realized. 

.30  =  'f^  to  be  gained.  .25  =  <^  of  1st  discount. 

72  =  gain  to  be  insured.  .75  =  ^  of  price  to  be  received  in. 

240  =  cost.  order  to  gain  30^  and  allow 

$"312  =  price  to  be  received.  ^^^  discount. 

.75  )  $312.00  $1.00  =  ^  of  price. 

.  .20  =  fo  of  2d  discount. 

$410  =  asking  price  m  order  

to  pay  $240,  gain  .80  =  ^  oi  price  to  be  received  in 
30,^,  and  allow  a  dis-  order  to  gain  30,*^  and  dis- 
count of  25^.  count  25^  and  20^. 

.80  )  $416.00  $1.00  ^  ^  of  price. 

,  .           .      .         T                   .20  —  ^  of  3d  discount. 
$520  =  asking  price  in  order  to        

pay  $240,  gain  30^,  and  .80  =  ^  of  price  to  be  received  in 

discount  25^c  and  20^.  order  to  gain  'dWo  and  dis- 


count 25^,  20^,  and  20^. 


.80  )  $520.00 


$550  =  asking  price  in  order  to  gain  30^  and 
allow  the  full  discount  series. 

Rule. — Add  to  the  cost  the  gain  required,  and  divide  consecutively  bjf 
1  minus  each  of  the  rates  in  the  discount  series. 

EXAMPLES   FOK   PRACTICE. 

577.  1.  What  must  be  the  asking  price  of  a  watch,  costing  $18,  that  33^^ 
may  be  gained,  after  allowing  the  purchaser  a  discount  of  20^  ? 

2.  Having  bought  an  invoice  of  lawn  mowers  at  $15  each,  I  desire  to  so  mark 
them  that  I  may  gain  20^,  and  still  discount  25^  and  20^  to  my  customers.  At 
what  price  must  each  be  marked  ? 

3.  Having  paid  $8800  for  a  stock  of  goods,  what  price  must  be  asked  for  it, 
in  order  to  gain  $1100  and  allow  12^^^^  and  10;^^  discount  ? 

Jf.  After  buying  velvet  at  $5  per  yard,  I  so  marked  it  as  to  allow  discounts  of 
25^,  20^,  and  16f^  from  the  marked  price,  and  yet  so  sell  it  as  to  lose  but  10^ 
on  my  purchase.     At  what  price  per  yard  was  the  velvet  marked  ? 

5.  The  cost  of  manufacturing  silk  hats  being  $36  per  dozen,  how  must  they 
be  marked,  that  a  gam  of  16f ^  may  be  realized  by  the  manufacturer,  after  allow- 
ing discounts  to  the  trade  of  20^  and  12^^^? 

6.  If  a  carriage  be  marked  33^j^  above  cost,  what  per  cent,  of  discount  can 
be  allowed  from  the  marked  price  and  realize  cost? 

7.  If  the  list  price  of  an  article  is  25^  advance  on  the  cost,  what  other  per 
cent,  of  discount  than  10^  must  be  allowed,  to  net  10^  gain  by  sale  ? 


186  EXAMPLES   IN   TRADE   DISCOUNT. 

8.  A  merchant  purchasing  a  bill  of  goods  was  allowed  discounts  from  the  list 
price  of  Ib-^,  10^,  10^,  and  fi^.  If  the  total  discount  allowed  was  $352.81, 
what  must  have  been  the  asking  price  of  the  goods  ? 

578.  To  find  a  Single  Equivalent  Per  Cent,  of  Discount,  a  Discount  Series 
being  given. 

Example. — What  single  rate  of  discount  is  equal  to  the  series  25^,  20^  lOji^, 
and  b'^c  ? 

Operation. 
IIOOO  =  assumed  list  price  or  base. 
250  =  1st  discount. 

$750  =  1st  rem.  or  2d  base. 

150  =  2d  discount  Explanation.  —  Assume  $1000  as  the  list 

'  price,  and  successively  deduct  the  discounts  as 

$600  =  2d  rem.  or  3d  base.  by  the  series,  and  compare  the  result  with  the 

60  =■  3d  discount.  base  assumed. 

$540  =  3d  rem.  or  4th  base. 

27  =  4th  discount. 
$513  =  4th  or  last  rem.  or  net  price. 

$1000  =  list  price,  or  base. 
513  =  net  price. 
$487  =  total  discount  on  $1000,  which,  divided  by  1000,  gives  48y'L,  the  per 
cent,  of  discount  equivalent  to  the  given  series. 

Rule. — From  $1000  as  a  list  pi'ice,  or  base,  take  the  discounts  in  order; 
subtract  the  final  remainder  from  the  hase  taken,  and  the  result  irill  be 
the  total  discount ;  then  point  off  from,  its  right  three  places  for  decimals, 
and  the  expression  thus  obtained  mill  be  the  equivalent  per  cent,  of  dis- 
count required. 

Remark. — This  is  the  usual  method,  and  it  is  more  convenient  for  business  men  than  to 
compute  the  net  price  for  each  sale  through  a  series  of  discounts. 

EXAMPLES  rOK  PRACTICE. 

579.     1.     Find  a  single  discount  equivalent  to  a  series  of  10^  and  10^. 

2.  Find  a  single  discount  equivalent  to  a  series  of  25^,  15^,  and  5^. 

3.  Find  a  single  discount  equivalent  to  a  series  of  30^,  20^,  lOj^,  and  3^^. 

Jf.  Goods  were  sold  25^,  35^,  20^,  and  Ib^  off  ;  what  single  discount  would 
have  insured  the  same  net  price  ? 

6.  What  is  the  difference  between  a  single  discount  of  50^  and  a  series  of 
20^,  20^,  and  10^  ? 

6.  What  per  cent,  of  the  list  price  will  be  obtained  for  goods  sold  at  a 
discount  of  35^,  20^,  15^,  10^  and  b'^  ? 

7.  From  a  list  price,  I  discounted  30^  25^  20^,  15^,  124^,  10^  and  b^. 
What  per  cent,  better  for  the  purchaser  would  a  single  discount  of  75^  have 
been  ? 


STOKAGE.  187 


STORAGE. 

580.  Storage  is  a  provision  mude  for  keeping  goods  in  a  wareliouse  for 
a  time  agreed  upon,  or  for  an  indefinite  time,  subject  to  accepted  conditions. 

581.  The  term  storage  is  used  also  to  designate  the  charges  for  keeping 
the  goods  stored. 

582.  Rates  of  storage  may  be  fixed  by  agreement  of  the  parties  to  tlie  con- 
tract, but  are  often  regulated  by  Boards  of  Trade,  Chambers  of  Commerce, 
Associations  of  Warehousemen,  and  by  legislative  enactment. 

583.  Storage  Charges  may  be  made  at  a  fixed  price  per  package  or  bushel, 
or  at  a  fixed  sum  for  a  term  or  terms ;  they  may  be  made  for  a  term  of  days  or 
months  ;  but  usually,  if  the  goods  stoz-ed  are  taken  out  before  the  storage  time 
expires,  the  charge  made  is  for  the  full  time. 

584.  The  rates  of  storage  often  vary  for  grains,  or  goods  of  different  grades 
or  values,  and  also  on  account  of  different  modes  of  shipment. 

Remarks. — Storage  Receipts,  especially  of  grains,  are  frequently  bought  and  sold  under  the 
name  of  "Warehouse  Receipts  "or  "Elevator  Receipts,"  as  representing  so  much  value  by 
current  market  reports. 

585.  Cash  Storage  is  a  term  applied  to  cases  in  which  the  payment  of 
charges  is  made  on  eaeii  withdrawal  or  shipment,  at  the  time  of  such  withdrawal 
or  shipment,  notwithstanding  the  fact  that  the  owner  may  still  have  goods  of 
the  same  kind  in  store  at  the  Avarehouse. 

586.  Credit  Storage  is  a  term  applied  to  cases  in  which  sundry  deposits  or 
consignments  are  received,  from  which  sundry  withdrawals  or  shipments  are 
made,  and  all  charges  adjusted  at  the  time  of  final  withdrawal. 

Remarks. — 1.  When  deposits  or  consignments,  and  withdrawals  or  shipments,  are  made 
at  different  times,  credit  is  to  be  given  for  the  amount  of  each  deposit  or  consignment,  from 
tlie  date  to  its  final  withdrawal  or  shipment,  and  credit  given  to  the  owner  or  consignor  for  each 
withdrawal  or  shipment,  from  date  up  to  the  time  of  settlement. 

2.  In  the  private  bonded  warehouses  of  the  United  States,  goods  may  be  taken  out  at  any 
time,  in  quantities  not  less  than  an  entire  package,  or,  if  in  bulk  of  not  less  than  1  ton,  by  the 
payment  of  duties,  storage,  and  labor  charges.  The  storage  charges  are  computed  for  periods 
of  one  month  each,  a  fractional  part  of  a  month  being  counted  the  same  as  a  full  month. 

3.  Drovers  sometimes  hire  cattle  fed  on  account,  entering  and  withdrawing  them  as  circum- 
stances require;  such  accounts  are  closed  in  the  same  manner  as  are  those  for  storage. 

587.  To  find  the  Simple  Average  Cash  Storage. 

Example. — There  was  received  at  a  storage  warehouse  :  Oct.  11,  300  bar. 
apples;  Oct.  30,  250  bar.  potatoes;  Nov.  13,  200  bar.  apples;  Nov.  20,  60  bar. 
quinces ;  Nov.  28,  280  bar.  apples.     The  merchandise  was  all  delivered  Dec.  2. 


188  STORAGE. 

If  the  contract  specified  that  the  rate  of  storage  was  5'/  per  barrel  for  a  period 
of  30  days  average  storage,  what  Avas  the  storage  bill  ? 

0PERA.TI0N. 

The  storage  of  300  bar.  for  52  da.  =  the  storage  of  1  bar.  for  loGOO  da. 
The  storage  of  250  bar.  for  33  da.  =  the  storage  of  1  bar.  for  8250  da. 
The  storage  of  200  bar.  for  19  da.  =  the  storage  of  1  bar.  for  3800  da. 
The  storage  of  60  bar.  for  12  da.  =  the  storage  of  1  bar.  for  720  da. 
The  storage  of  280  bar.  for    4  da.  —  the  storage  of  1  bar.  for    1120  da. 


The  total  storage  =  the  storage  of  1  bar.  for  29490  da. 
Or,  983  periods  of  30  days  each;  $.05  x  983  =  1^49.15,  storage  bill. 

Explanation. — The  300  barrels  constituting  the  first  deposit  or  delivery  were  stored  from 
Oct.  11  to  Dec.  2,  or  for  52  days;  the  storage  of  300  bar.  for  52  days  equals  the  storage  of  1 
barrel  for  15600  days;  the  storage  of  250  barrels  for  33  daj's  equals  the  storage  of  1  barrel  for 
8250  days;  that  of  200  barrels  for  19  days  equals  1  barrel  for  3S00  days;  that  of  60  barrels  for 
12  days  equals  1  barrel  for  720  days;  that  of  280  barrels  for  4  days  equals  1  barrel  for  1120 
days.  The  total  storage  was  equal  to  that  of  1  barrel  for  29490  days,  or  for  983  storage  terms 
or  periods  of  30  days  each.  Since  the  storage  charge  was  5^^  per  barrel  for  each  average  period 
of  30  days,  the  charge  would  amount  to  |.05  X  983,  or  $49.15. 

Rule. — Multiply  the  nuniber  of  aiUcles  of  each-  receipt  by  the  iiuiriber 
of  days  between  the  time  of  their  deposit  and  withdrawal;  divide  the 
Slim  of  these  products  hy  the  number  of  days  in  the  storage  period,  and 
midtiply  the  qiiotieiit  by  the  charge  per  period. 

KXAMPLE.S   FOR   PRACTICE. 

688.  1.  There  was  received  at  a  warehouse:  May  30,  4000  bu.  wheat;  Jitiie 
5,  2600  bu.  oats;  June  24,  3500  bu.  barley;  July  18,  5000  bu.  corn.  If  all  of  this 
was  shipped  July  20,  wlutt  was  the  storage  bill,  the  charge  being  14^-  per  bushel 
per  term  of  30  days  average  storage  ? 

2.  A  farmer  received  for  pasture:  April  30,  12  head  of  cattle;  May  15,  14 
head  of  cattle;  May  23,  27  head  of  cattle;  June  9,  5  head  of  cattle;  June  30,  8 
head  of  cattle;  July  16,  40  head  of  cattle.  All  were  delivered  July  25,  and  the 
charges  were  75^  per  head  for  each  week  of  7  days  average  pasturage.  How 
mucl>  was  his  bill  ? 

589.    To  find  the  Charge  for  Storage  with  Credits. 


Example. — The  storage  charges  being  ■2<f:  per  barrel  for  a  month  of  30  days 
average,  what  will  be  the  bill  in  the  following  transaction  ? 

Received. 
July  19,  100  bar.;  July  31,  240  bar.: 


Sept.  8,  3G0  bar. 


Delivered. 
Aug.  15,  300  bar. ;  Sept.  12,  2()0  bar.; 
Oct.  1,  200  bar. 


EXAMPLES    IX   STORAGE.  189 

Operation. 

Prom  July  19  to  July  31  =  12  da.;  100  bar.  stored  for  12  da.  =  1  bar.  stored  for     1200  da. 
July  31  240  bar.  received. 

From  July  31  to  Aug.  15  =  15  da.;  340  bar.  stored  for  15  da.  =  1  bar.  stored  for     5100  da. 
Aug.  15  300  bar.  delivered. 

From  Aug.  15  to  Sept.  8  =  24  da. ;     40  bar.  remaining;  for  24  da.  =  1  l)ar.  stored  for   960  da. 
Sept.  8  360  bar.  received. 

From  Sept.  8  to  Sept.  12=4  da. ;  400  bar.  stored  for  4  da.  =  1  bar.  stored  for       1600  da. 
Sept.  12  200  bar.  delivered. 

From  Sept.  12  to  Oct.  1  =  19  da. ;  200  bar.  remaining  for  19  da.  =  1  bar.  stored  for  3800  da. 
Oct.  1  200  bar.  delivered. 


000  Total  =  1  bar.  stored  for  12660  da.. 

Or,  422  terms  of  30  da.  each;  $.02  X  422  =  $8.44,  total  storage  bill. 

Explanation. — 100  barrels  were  stored  for  12  days,  when  240  barrels  were  added;  these  340 
barrels  were  stored  15  days,  when  300  barrels  were  withdrawn;  the  40  remaining  barrels  were 
stored  24  days,  when  360  barrels  were  added;  these  400  barrels  were  stored  4  days,  when  200 
barrels  were  withdrawn;  the  remaining  200  barrels  were  stored  19  days  and  then  withdrawn. 
The  total  storage  thus  equalled  that  of  1  barrel  for  12660  days,  or  for  422  terras  of  30  days 
€ach;  and  since  the  charge  for  1  term  is  $  .02,  for  422  terms  it  would  be  422  times  $  .02,  or 
$8.44,  the  total  amount  of  the  bill. 

Rule.— I.  Multiply  the  ninnher  of  articles  first  received  hy  the  nurti- 
ber  of  days  between  the  date  of  their  receipt  and  the  date  of  the  next 
receipt  or  delivery ;  add  the  number  of  articles  of  such  next  receipt,  or 
subtract  the  iiuniber  of  sucJo  delivery,  as  the  case  may  be,  and  so  pro- 
ceed to  the  time  of  final  delivery. 

II— Divide  the  aggregate  storage  by  the  number  of  days  in  tlie  storage 
term,  and  multiply  the  quotient  by  the  storage  charge  per  term. 

EXAMPLES   FOK   PKACTICE. 

590.  1.  What  will  be  the  storage  charge,  at  4|{?!'  per  barrel,  for  a  term  of 
thirty  days  average,  in  the  following  transaction  ? 

Delivered. 
Mar.    1,     100  bar.  apples. 
Mar.  28,     190  bar.  flour. 
Apr.  15,       60  bar.  potatoes. 
Apr.   "         60  bar.  flour. 
Apr.  29,     230  bar.  flour. 

2.  A  drover  hired  pasture  of  a  farmer,  agreeing  to  pay  84.20  per  head  of 
stock  pastured  for  each  average  term  of  30  days.  What  was  the  amount  of  the 
bill,  the  receipts  and  deliveries  being  as  follows  ? 

Received.  Delivered. 


Feb. 

8, 

iieceiveu. 
180  bar. 

flour. 

Feb. 

27, 

100  bar. 

api)les. 

Mar. 

8, 

60   bar. 

potatoes. 

Mar. 

13, 

300  bar. 

flour. 

June  15,  21  head  of  cattle. 

June  27,  20  head  of  cattle. 

July     5,  15  head  of  cattle. 

July  29,  40  head  of  cattle. 

July  31,  40  head  of  cattle. 


July    1,  30  head  of  cattle. 

July  20,  15  head  of  cattle. 

July  30,  15  head  of  cattle. 

Aug.  21,  the  remainder. 


190 


EXAMPLES    IN   STORAGE. 


591.     To  find  the  Storage  where  Charges  Vary. 

Example. — At  a  warehouse  there  was  received  and  delivered  flour,  as  follows: 


Delivered. 
Jan.  23,     250  bar. 
1,     400  ])ar. 


Mar. 


Received. 

Jan.    3,     150  bar. 

Jan.  20,     200  bar. 

Feb.     1,     300  bar. 

The  storage  charge  on  the  above  was,  bf-  per  barrel  for  the  first  10  dajs  or 
part  thereof,  and  3^  per  barrel  for  each  subsequent  period  of  10  days  or  part 
thereof.     What  sum  must  be  paid  in  settlement? 


Operation. 


Date.  Receipts  and  Deliteries. 

Jan.    3,    received  150  l)ar. 
200     •' 


20,  ''  __ 

350 
23,  delivered  250 


Feb. 


Mar. 


100 
1,    received  300 

400 
1,  delivered  400 


in  store. 

150  bar.  stored  20  davs,  or 

100         '•  3     '' 

remainder. 


Rate.        Storage. 


terms,  89*  =  ^12.00 
term,   5'/  =      5.00 


in  store. 

100  bar.  stored  40  davs,  or  4  terms,  14^  =  $14.00 

300         "      -      28     •'  3       *•       11^  =      5.00 


Total  storage. 


=  $64.00 


Explanation. — Of  the  250  barrels  delivered  Jan.  23,  150  barrels  had  been  in  store  since 
Jan.  3,  20  days  or  2  terms,  and  the  charge  was  5  cents  plus  3  cents,  or  8  cents  per  barrel, 
which  equals  $12  storage.  The  remaining  100  barrels  of  the  delivery  of  Jan.  23,  had  been 
in  store  only  since  Jan.  20,  3  days  or  1  term,  at  0  cents  per  barrel,  equal  to  f  0  storage.  Of  the 
400  barrels  delivered  Mar.  1,  100  barrels  had  been  in  store  since  Jan.  20,  40  days  or  4  terms, 
at  5  cents  plus  3  cents  plus  3  cents  plus  3  cents,  or  14  cents  per  barrel,  equal  to  $14  storage; 
while  the  remaining  300  barrels  had  been  in  store  since  Feb.  1,  28  days  or  3  terms,  at  5  cents 
plus  3  cents  plus  3  cents,  or  11  cents  per  barrel,  equal  to  $33  storage.  By  addition,  the  total 
storage  is  found  to  be  $64. 

l^wXe.—Mujltiply  the  number  of  articles  of  each  delivery  by  tJie  charge 
fur  ihe  term  or  imns  stored,  and  add  the  products  so  obtained. 


EXAMPLK   FOK   PK.4.CTICE. 


592. 

follows: 


1.     The  receipts  and  deliveries  of  goods  at  a  storage  warehouse  were  as 

Received. 
Sept.     2,     100  bar. 


Sept. 
Oct. 
Oct. 
Nov. 


25, 
19, 
31, 

7, 


200  bar. 

350  bar. 

150  bar. 

200  bar. 


Deli  I 

'ered. 

Sept. 

20, 

100  bar. 

Sept. 

30, 

100  bar. 

Oct. 

10, 

100  bar. 

Oct. 

20, 

100  bar. 

Oct. 

30, 

100  bar. 

Nov.  20,     the  remainder. 
The  contract  required  the  payment  of  0'/  per  barrel  for  the  present  term  of 
30  days  or  fraction  thereof,  and  3^  per  barrel  for  each  subsequent  term  of  30 
days  or  fraction  thereof.     Find  the  storage  bill. 


COMMISSION.  191 


COMMISSION. 

593.  An  Agent  is  a  person  who  transacts  business  for  another;  as,  the 
purcliase  or  sale  of  merchandise  or  real  estate,  collecting  or  investing  money,  etc. 

594.  An  agent  who  receives  goods  to  be  sold  is  sometimes  called  a  factor  or 
commission  merchant;  one  employed  to  buy  or  sell  stocks  or  bonds,  or  to  nego- 
tiate money  securities,  is  called  a  hroker. 

595.  Commission  is  an  allowance  made  to  agents  or  commission  merchants 
for  transacting  business.  It  is  usually  a  percentage  of  the  money  involved  in 
the  transaction,  although  sometimes  it  is  computed  at  a  certain  price  2)er  bale, 
bushel,  barrel,  etc. 

596.  The  Agent's  Commission  for  selling  is  computed  on  the  gross  pro- 
ceeds, and  for  purchasing  on  tlie  prime  cost. 

597.  The  Principal  is  the  person  for  whom  the  business  is  transacted. 

598.  A  Consignment  is  a  shipment  of  goods  from  one  party  to  another,  to 
be  sold  on  account  of  the  shipper,  or  on  joint  account  of  the  shipper  and  the 
consignee.  The  shipper  is  called  the  Consignor,  and  the  one  to  whom  the  goods 
are  shipped  is  called  the  Consignee. 

599.  Ouaranty  is  a  per  cent,  charged  by  an  agent  for  assuming  the  risk  of 
loss  from  sales  made  by  him  on  credit,  or  for  giving  a  pledge  of  tlie  grade  of  goods 
bought;  it  is  computed  the  same  as  are  commission  charges. 

600.  The  Gross  Proceeds  of  a  sale  or  collection  is  the  total  amount 
received  by  the  agent  before  deducting  commission  or  other  charges. 

601.  The  Net  Proceeds  is  what  remains  after  all  charges  have  been 
deducted. 

Remakks. — Charges  maybe  for  commission,  guaranty,  freight,  inspection,  cartage,  storage, 
or  any  other  outhiy  incident  to  the  sale. 

602.  All  Account  Sales  is  a  statement  in  detail  rendered  by  a  Consignee  to 
his  Consignor,  showing  the  sales  of  the  consignment,  all  of  the  charges  or 
expenses  attending  the  same,  and  the  net  j^roceeds. 

603.  All  Account  Purchase  is  a  detailed  statement  made  by  a  purchasing 
agent  to  his  princijial,  having  the  quantity,  grade  and  i)rice  of  goods  bought  on 
his  account,  all  the  expenses  incident  to  the  purchase,  and  the  gross  amount  of 
the  purchase. 

60'1.     Commission  compares  with  Abstract  Perce?itage,  as  follows: 
The  Prime  Cost  or  Gross  Selling  Price  =  Base. 
The  Rate  Per  Cent,  of  Commission  =  Kate. 


lO'J  COMMISSION. 

The  Commission  for  either  buying  or  selling,  or  for  guaranty  of  quality  or 
credit  =  Percentage. 

The  remittance  to  Purchasing  Agent,  including  both  Commission  and  Invest- 
ment =  Amount. 

The  Selling  Price,  minus  the  Commission  =  Difference. 

605 . — To  find  the  Commission,  the  Cost  or  Selling  Price  and  Per  Cent,  of 
Commission  being  given. 

Example:. — IIow  much  commission  will  be  due  an  agent  Avho  buys  $8000 
worth  of  coal,  on  a  commission  of  o'i? 

^^^^■^^^°^*  Explanation.— Since  the  rate  of  commission 

#800(1  =  investment  or  base.  is  .5  per  cent.,  the  whole  commission  due  the  agent 

.05  =  per   cent,  of  commission.        will  be  5  per  cent,  of  the  investment,  $8000,  or 
$.100  =  commission  or  percentage.        $400. 
Remark. — In  case  of  sales,  proceed  in  like  manner,  treating  the  selling  price  as  the  base. 

^\l\e.—JIiiJf7phj  the  cost  or  selling  price  hij  the  rate  per  cent,  of 
eoDiniissioji. 

Formula. — Commission  =  Cost  or  Selling  Price  X  Rate  per  cent,  of  Commission. 

EXAMPLES   FOK   PRACTICE. 

606.  i.  A}i  agent  sold  a  house  and  lot  for  «6000,  and  charged  3^  for  his 
services.     How  much  was  the  commission? 

2.  Having  agreed  to  pay  an  agent  3f^  for  all  purchases  made  by  him,  how 
much  will  be  due  him,  if  he  buys  for  me  goods  costing  $2500? 

3.  If  an  agent's  charges  are  2f^,  how  much  commission  will  he  earn  by  selling 
property  valued  at  $12500? 

Jf.  I  owned  one-half  of  a  stock  of  goods  sold  by  an  agent  for  $10000.  If  the 
agent  charged  b\'i  for  selling,  how  much  commission  must  I  pay? 

5.  An  auctioneer  sold  a  store  for  $8500,  and  its  contents  for  $7350.  How 
much  did  his  fees  amount  to,  at  If;^? 

6.  A  real  estate  agent  sold  a  farm  oi  91  acres,  at  $120  per  acre,  on  a  com- 
mission of  2<;  and  the  stock  and  utensils  on  the  farm  for  $3150,  on  a  commission 
of  5<.     What  was  the  amount  of  his  commission? 

607.  To  find  the  Investment  or  Gross  Sales,  the  Commission  and  Per  Cent 
of  Commission  being  given. 

Example. — If  uu  agent's  rate  of  commission  is  2f(,  what  value  of  goods  must 
he  sell  to  earn  a  commission  of  $50? 

Operatiox.  Explanation. — Since  the  agent's  commission  is  2  per 

qqqj  cent.,  he  earns  2  cents  by  selling  $1  worth  of  goods;  the 

2^  =   02  ^  $50  00  value   of    the  goods  sold,   therefore,  must  be  as    many 

— — —  times  $1  as  2  cents  is  contained  times  in  $50;  2  cents  is 

$-.500  gross  sales,     contained  in  $50,  2500  times,  and  2500  times  $1  is  $2500. 

Remark. — T^'hen  commission  for  purchase  is  given  and  cost  required,  proceed  in  like 
manner. 


EXAMPLES   IX    COMMISSION.  193 

Uule.— Divide  the  commission  by  the  rate  per  cent,  of  commission. 

Formula. — Prime  Cost  or  Gross  Selling  Price  =  Commission,  divided  by  the 
Eate  Per  Cent,  of  Commission. 

EXAMPLES  FOK    PRACTICE. 

608.  1.  What  amount  of  merchandise  must  be  purchased  on  a  commission 
•of  Z\<j(>,  in  order  that  an  agent  may  receive  a  commission  of  $175? 

2.  An  agent  received  l>306.25  for  selling  wheat,  on  a  commission  of  \\^. 
What  Avas  the  amount  of  the  sales? 

3.  A  collector's  charges  of  5^  for  collecting  a  note  amounted  to  $14. 10.  What 
sum  was  collected? 

4.  A  factor  charged  $216.80  for  selling  a  consignment  of  canned  fruit.  If 
his  commission  was  2|^,  what  must  have  been  the  gross  sales? 

5.  I  paid  a  grain  dealer  \\'/o  for  buying  corn  for  me,  at  62^  per  bushel.  If  his 
•commission  amounted  to  $83.70,  how  many  bushels  did  he  buy? 

6.  A  Mobile  factor  earned  $99.75  by  selling  cotton,  at  2f^  commission.  How 
many  bales,  averaging  560  lb.,  did  he  sell,  the  price  being  15^  per  pound  ? 

6(M).  To  find  the  Investment  and  Commission,  when  Both  are  Included  in  a 
Remittance  by  the  Principal. 

Example. — If  $1050  is  sent  to  a  Saginaw  agent  for  the  purchase  of  salt,  how 
much  will  he  invest,  his  rate  of  commission  being  o,'^? 

Operation.  Explanation.— For  each  dollar  invested 

^1.00  =  investment.  by  the  agent,  the  principal  supplies  the  dollar 

.05  =  commission.  invested  and  5  cents  for  the  agent's  services; 

a,-,  AK  ^i-,    ^        i.  i.         ■      •      ^     J!         1  therefore  the  agent  will  invest  only  as  manv 

$1.05  =  actual  cost  to  principal  of  each  ,  „       .       ,.       *^    ,      ^  ^.  ^.  . 

,  ,  dollars  in  salt  as  $1  plus  5  cents,  or  $1.05,  is 

dollar  invested  by  agent.  contained  times  in  the  remittance,  $1050;  1.05 

1.05  )  $1050.00  is  contained  in  $1050.  1000  times;  hence  the 

$1000  sum  invested  in  salt.  investment  is  $1000. 

Rule.— Divide  the  remittance  by  1  plus  the  rate  per  cent,  of  coinmission. 

Remarks. — 1.  All  computations  in  commission  may  be  made  by  applying  the  principles  of 
Percentage. 

2.  When  a  charge  is  made  for  guaranty,  add  the  per  cent,  of  guaranty  to  1  plus  the  rate  per 
■cent,  of  commission,  and  proceed  as  above. 

Formula. — Investment  =  Remittance  to  Agent  -4-  1  plus  the  Rate  Per  Cent, 
■of  Commission. 

EXAMPI.es  FOK  PRACTICE, 

610.  1.  An  agent  receives  $12504.20,  with  instructions  to  invest  in  avooI. 
If  his  commission  is  3c^,  how  many  dollars  worth  of  wool  will  he  purchase? 

2.  How  many  pounds  of  wool,  at  27{^  per  pound,  can  be  bought  for  $8424,  if 
the  agent  is  allowed  Afo  for  purchasing? 

8.     I  remitted  $1306.45  to  a  Boston  agent  for  the  purchase  of  soft  hats.     If 
the  agent's  commission  is  4,'^,  and  he  makes  an  added  charge  of  2^  for  guaranty 
of  quality,  how  many  dozen  hats,  at  $8.50  per  dozen,  should  he  send  me? 
13 


194  EXAMPLES   IX   COMMISSION. 

4.  An  agent  receivies  $13760.80  to  invest  in  land,  after  deducting  his  charges 
of  3^.     What  amount  of  commission  will  he  receive? 

5.  A  real  estate  agent,  whose  stated  commission  is  2^^,  receives  38302.50  to 
invest  in  Iowa  prairie,  at  $5,40  per  acre.  How  many  acres  did  he  jnirchase,  and 
and  how  much  Avas  his  commission. 

6.  I  remitted  $300  to  an  agent  for  the  purchase  of  hojis.  If  the  agent's 
charges  were  h'fo  for  purchase  and  $6  for  inspection,  how  many  pounds,  at  10^ 
per  pound,  ought  he  to  buy? 

MISCEL,I.AXEOrS   EXAMPLES. 

611.  1.  A  collector  obtained  75*^  of  the  amount  of  an  account,  and  after 
deducting  12'?;  for  fees,  remitted  his  principal  $495.  What  was  the  amount  of 
his  commission? 

2.  A  Hartford  fruit  dealer  sent  a  Lockport  agent  $1946.70,  and  instructed 
him  to  buy  apples  at  $1.40  per  barrel.  The  agent  charged  3fo  for  buying,  and 
shipped  the  purchase  to  his  principal  in  six  car  loads  of  an  equal  number  of 
barrels.     How  many  barrels  did  each  car  contain? 

3.  Find  the  per  cent,  of  commission  on  'a  purchase,  if  the  gross  cost  is 
$2048.51,  the  commission  $87.30,  the  cartage  820.  and  other  charges  $1.21. 

4.  11500  bushels  of  wheat  were  bought  through  an  agent,  Avho  charged  \f^ 
for  buying.  If  the  agent  paid  85{#  per  bushel  for  the  wheat,  $762.50  freight, 
and  $12.50  insurance,  what  sum  should  be  remitted  to  him  in  full  settlement? 

5.  A  collector  obtained  75;:?;  of  a  doubtful  account  amounting  to  S1750.  How 
much  was  his  per  cent,  of  commission,  if,  by  agreement  with  the  principal,  the 
commission  was  to  be  50,*^  of  the  net  proceeds  remitted? 

6.  A  farmer  received  from  his  city  agent  $490  as  the  net  proceeds  of  a  ship- 
ment of  butter.  If  the  agent's  commission  is  3ff,  delivery  charges  $6.80,  and  b^^^ 
charge  is  made  for  guaranty  of  quality  to  purchasers,  how  many  pounds,  at  27^ 
per  pound,  must  have  been  sold,  and  how  much  commission  was  allowed? 

7.  An  agent  sold  2000  bu.  Alsike  clover  seed,  at  $7.85  per  bushel,  on  a  com- 
mission of  5^;  and  1200  bu.  medium  red,  at  $5.20  per  bushel,  on  a  commission 
of  ^\ic\  taking  the  purchasers  3  month's  note  for  the  amount  of  the  sales.  If 
the  agent  charges  4<  for  his  guaranty  of  the  notes,  what  amount  does  he  earn  by 
the  transaction? 

8.  An  agent  bought  l)utter  on  a  commission  of  10^,  cheese  on  a  commission 
of  6^,  and  eggs  on  a  commission  of  h'lr.  If  his  commission  for  buying  the  butter 
was  $21,  for  buying  the  cheese  $21.60,  and  for  buying  the  eggs  |22,  and  he 
charges  25^  additional  for  guaranteeing  the  freshness  of  the  eggs,  what  sum 
should  the  jjrincipal  remit  to  i)ay  for  purchases  and  charges? 

9.  Find  the  pet  proceeds  of  a  sale  made  by  an  agent  charging  Z^i^,  if  inci- 
dental charges  and  commission  charges  were  each  $41.30. 

10.  From  a  consignment  of  3160  jjounds  of  tea,  sold  by  an  agent  at  30^  per 
pound,  the  consignor  received  as  net  proceeds  $853.74.  What  was  the  per  cent, 
of  commission  charged  for  selling,  if  the  charges  for  storage  and  insurance 
amounted  to  $51.60? 


EXAMPLES   IN"   COMMISSION.  105 

11.  Find  the  gross  jiroceeds  of  a  sale  made  by  an  agent  charging  2^^  for  com- 
mission, hi  for  guaranty,  $17.G5  for  cartage,  $11.40  for  storage,  and  *3.25  for 
insurance,  if  the  net  proceeds  remitted  amount  to  $1714.10. 

12.  A  Milwaukee  agent  received  $83195.28,  with  instructions  to  invest  one- 
half  of  it  in  wheat,  at  80^'  per  bushel,  and  the  balance,  less  all  commissions,  in 
wool,  at  20^  per  pound.  If  his  commission  for  buying  the  wheat  is  2,<,  and  that 
for  buying  the  wool  is  5^,  how  many  pounds  of  avooI  Avill  he  buy,  and  Avhat  Avill 
be  the  amount  of  his  commissions? 

13.  I  sent  $3402.77  to  my  Atlanta  agent  for  the  purchase  of  sweet  i)otatoes, 
at  $1.60  per  barrel;  his  charges  were,  for  commission,  2^^;  guaranty,  3,<;  dray- 
age,  1^  per  barrel;  and  freight,  $200.  How  many  barrels  did  he  buy,  and  how 
much  unexpended  money  was  left  in  his  hands  to  my  credit? 

11^.  A  Texas  buyer  shipped  33000  lb.  of  coarse  wool  to  a  Boston  agent  to  Ijo 
sold  on  commission,  and  gave  instructions  for  the  net  proceeds  to  be  invested  in 
leather.  If  the  agent  sold  the  wool  at  18^/  per  pound,  on  a  commission  of  2f^, 
and  charged  10^  for  the  purchase  and  guaranty  of  grade  of  the  leather,  what  a\  as 
the  amount  of  his  commissions? 

i'5.  I  receiA'Cd  from  Duluth  a  cargo  of  IGOOO  bu.  of  wheat,  which  I  sold  at 
$1. 10  per  bushel,  on  a  commission  of  4^6^;  by  the  consignor's  instructions  I  invested 
the  net  proceeds  in  a  hardware  stock,  for  Avhich  I  charged  5;^  commission.  What 
was  the  total  commission,  and  how  much  was  invested  in-iiardware? 

16.  Having  sent  a  New  Orleans  agent' $1835. 40  to  be  invested  in  sugar,  after 
allowing  3f^  on  the  investment  for  his  commission,  I  received  32400  pounds  of 
sugar.     "What  price  per  i)ound  did  it  cost  the  agent? 

n.  An  agent  in  Providence  received  $828  to  invest  in  prints,  after  deducting 
his  commission  of  Z^'fo.  If  lie  paid  74-^'-  per  yard  for  the  prints,  how  many  yards 
did  he  buy? 

18.  The  fees  of  the  general  agent  of  an  insurance  comjjany  arc  h'^  on  all  sums 
received,  and  5^  additional  on  all  sums  renuiiuing  in  his  hands  at  the  end  of  the 
year,  after  all  losses  and  the  expenses  of  his  office  are  paid.  He  receiA^es  during 
the  year  $117410.25,  paid  losses  to  the  amount  of  $01140.50,  and  the  expenses 
of  his  office  Avere  $3207.70.     Find  his  total  fees. 

19.  An  agent  sold  on  commission  81  self-binders,  at  $140  each,  and  113 
mowers,  at  $05  each,  remitting  $10224.90  to  his  principal.  Find  the  rate  of 
commission. 

20.  A  commission  merchant  received  a  consignment  of  600  bales  of  cotton, 
of  an  average  weight  of  510  pounds,  Avhich  he  sold  at  124^  per  pound,  on  a 
commission  of  3^,  charging  10^'  per  bale  for  cartage.  He  invested  for  the  con- 
signor $9410.20  in  bacon,  charging  5^^  for  buying,  and  remitted  cash  to  balance 
consignor's  account.     Hoav  much  Avas  the  cash  remittance? 

21.  An  agent  received  $4325,  to  invest  in  mess-pork,  at  $10  \wx  barrel,  after 
deducting  his  i)urchasing  commission  of  A^.  If  the  charges  for  incidentals  were 
$81.40,  besides  cartage  of  75^/  per  load  of  8  barrels,  how  nuiny  barrels  did  he  buy, 
and  Avliat  unexpended  balance  does  he  place  to  the  credit  <!f  his  principal?  , 


106  MISCELLANEOUS    EXAMPLES    IN'    COMMISSION. 

22.  A  Street-ear  company  bought  35  horse-cars  through  a  Troy  agent,  at  $850 
each.  If  tlie  freight  was  $17.50  on  each  car,  and  the  agent's  commission  3i^^ 
for  purchasing,  what  was  tlie  total  cost  to  tlie  company? 

23.  I  received  from  Day  &  Son,  of  Chicago,  a  ship  load  of  corn,  which  I  sold 
for  eo^/'  jjer  bushel,  on  a  commission  of  4*^;  and,  by  the  shipper's  instructions, 
invested  the  net  proceeds  in  barley,  at  75^  per  bushel,  charging  b'i  for  buying; 
mv  total  commission  was  ^1350.  How  many  bushels  of  corn  did  Day  &  Son  ship, 
;ind  liow  many  bushels  of  barley  should  they  receive? 

;?4.  An  agent  sold,  on  commission,  1750  barrels  of  mess-pork,  at  $16.50  per 
barrel,  and  508  barrels  of  short-ribs,  at  $18  per  barrel,  charging  $112.50  for 
cartage,  and  $5,55  for  advertising.  He  then  remitted  to  his  principal  $36000, 
the  net  proceeds.     Find  the  rate  of  commission. 

25.  A  Wichita  dealer  sent  12  c.ir  loads  of  corn,  of  825  bushels  each,  to  an 
agent  in  Baltimore,  where  it  was  sold  at  62^{#  per  bushel,  on  a  commission  of  5fJ, 
the  agent  paying  $682.50  freight.  By  shippers  instructions,  the  agent  invested 
the  net  proceeds  m  a  hardware  stock,  charging  3^  for  buying.  How  much  was 
invested  in  liardware? 

26.  The  holder  of  a  doubtful  claim  of  $850,  handed  it  to  an  agent  for  collec- 
tion, agreeing  that,  for  every  dollar  sent  him  by  the  agent,  the  agent  might  keep 
for  himself  20^^  The  agent  succeeded  in  collecting  but  SOf^  of  the  debt.  How 
mucli  did  the  agent  remit,  how  much  commission  did  he  receive,  and  what  was 
his  per  cent,  of  commission? 

27.  I  remitted  $10500  to  a  Duluth  agent  to  be  invested  in  wheat,  allowing 
him  a  commission  of  3^^  for  investing.  The  agent  paid  95jZ^  per  bushel  for  the 
wheat,  and  charged  me  1\<P  a  bushel  per  month  for  storage.  At  the  end  of  4 
months  the  agent  sold  the  wheat  at  $1.10  per  bushel,  on  a  commission  of  5^. 
If  I  paid  $350  for  the  use  of  the  money,  did  I  gain  or  lose  by  the  operation,  and 
how  much? 

28.  My  Memphis  agent  sends  me  an  account  purchase  of  350  bales  of  cotton, 
averaging  480  pounds  each,  bought  at  15{#  per  pound,  on  a  commission  of  2^,^. 
His  charges,  other  than  for  commission,  Avere:  freight  advanced,  $120.50;  cartage, 
$53.25;  and  insurance,  $13.75.     "What  sum  should  I  remit  to  pay  the  account? 

29.  A  Charleston  factor  received  from  Cincinnati  a  consignment  of  corn, 
which  he  sold  at  75^  per  bushel,  on  a  commission  of  5<;  and  by  instructions  of 
the  consignor  invested  the  net  jiroceeds  in  cotton,  at  20^  per  pound,  charging 
Z^  for  buying,  and  3;^  additional  for  guaranty  of  quality.  If  the  total  amount 
of  the  agent's  commission  and  guaranty  was  $1640,  how  many  bushels  of  corn 
were  received? 

30.  A  Buffalo  brewer  remitted  $21500  to  a  Toronto  commission  merchant, 
with  instructions  to  invest  40^^  of  it  in  barley,  and  the  remainder,  less  all  charges, 
in  hops.  The  agent  paid  60^  per  bushel  for  barley,  and  200  i)er  pound  for  hops, 
charging  2^  for  buying  the  barley,  3,^  for  buying  the  hops,  and  5^  for  guaran- 
teeing the  quality  of  each  purchase.  If  his  incidental  charges  were  $187.50, 
what  quantity  of  each  product  did  he  buy,  and  what  was  the  amount  of  his 
commission? 


CUSTOM-HOUSE   BUSINESS.  197 


CUSTOM-HOUSE    BUSINESS. 

612.  Duties,  or  Customs,  are  taxes  levied  by  tlie  Goverumeut  on  imported 
goods,  for  revenne  purposes  and  for  the  protection  of  home  industry. 

613.  Duties  are  of  two  kinds,  ad  valorem  and  specific. 

614.  An  Ad  Yalorem  Duty  is  a  certain  per  cent,  assessed  or  levied  on  the 
actual  cost  of  the  goods  in  the  country  from  which  tliey  are  imported,  as  sliown 
by  the  invoice. 

615.  A  Specific  Duty  is  a  tax  assessed  or  levied  upon  the  number,  vreight, 
or  measure  of  goods,  regardless  of  their  value;  as,  a  fixed  sum  per  bale,  ton, 
barrel,  etc. 

Remark. — Upon  certain  goods  both  specific  and  ad  valorem  duties  are  levied. 

616.  A  Custom-House  is  an  office  established  by  the  Government  for  the 
transaction  of  business  relating  to  duties,  and  for  the  entry  and  clearance  of 
vessels. 

Remark. — 1.  The  ports  at  which  custom-houses  are  established  are  called  ports  of  entry. 

2.  The  waters  and  shores  of  the  United  States  are  divided  into  collection  districts,  in  each 
of  which  there  is  a  port  of  entry,  which  is  also  a  port  of  delivery;  other  ports  than  those  of 
entry  may  be  specified  as  porfo  of  delivery.  Duties  are  paid,  and  entries  and  clearances  made, 
at  ports  of  entry  only;  but  after  vessels  have  been  properly  entered,  their  cargoes  may  be 
discharged  at  any  port  of  delivery. 

617.  An  Invoice,  or  Manifest,  is  a  written  account  of  the  particular 
goods  sent  to  tlie  purchaser  or  factor,  with  the  actual  cost,  or  value,  of  such 
goods,  made  out  in  the  currency  of  the  country  from  which  they  are  imported. 

Remarks^ — All  invoices  are  made  out  in  the  weights,  measures,  etc.,  of  the  place  from  which 
the  goods  are  imported. 

618.  A  Tariff  is  a  schedule  of  goods,  and  tlic  rates  of  imjiort  duties  imposed 
by  law  (»ii  tlie  same. 

619.  'V\\v  Free  List  includes  classes  of  goods  that  are  exempt  from  duty. 

620.  Tonnage  is  a  tax  levied  upon  a  vessel  independent  of  its  cargo,  for  the 
privilege  of  coming  into  a  port  of  entry. 

621.  Duties  are  collected  at  the  port  of  entry  by  a  custom-house  officer 
appointed  by  tiie  United  States  Government,  and  known  as  tlie  Collector  of  the 
Port.     Under  him  are  deputy  collectors,  appraisers,  weighers,  gaugers,  etc. 

622.  The  Collector  of  the  Port  supervises  all  entries  and  papers  pertain- 
ing to  them;  estimates  all  duties,  receives  all  moneys,  and  employs  all  weighers, 
gaugers,  etc. 

62o.  Before  estimating  specific  duties,  allowjinces  are  made;  these  allowances 
are  called  Tare,  Leakage,  Breakage,  etc. 


198  CUSTOM-HOUSE   BUSINESS. 

6*24.  Tare  i-  an  iillf)wance  made  for  the  box,  bag,  crate,  or  other  covering  of 
the  goods. 

625.  Leakage^  determined  by  gauging,  is  an  allowance  made  for  waste  of 
liquid^^  imported  in  barrels  or  casks. 

6*26.     Breakage  is  an  allowance  made  for  loss  of  liquids  imported  in  bottles. 

627.  Gross  Weight  is  the  weight  before  any  allowances  are  made. 

628.  Xet  Weight  is  the  weight  after  all  allowances  have  been  made. 

Remarks. — 1.  The  ton  used  at  the  Tnited  States  Custom-Houses  is  of  2240  avoirdupois 
pounds. 

2.  Duties  are  not  computed  on  fractions  of  a  dollar;  if  the  cents  in  the  invoice  are  lesAthan 
50,  they  are  rejected;  if  50  or  more,  they  are  counted  as  a  dollar. 

629.  The  Naval  Officer,  appointed  only  at  the  more  important  ports, 
receives  copies  of  all  manifests,  countersigns  all  documents  issued  by  the  Col- 
lector, and  certifies  his  estimates  and  accounts. 

630.  The  Surveyor  superintends  the  employees  of  the  Collector,  and  revises 
entries  and  permits. 

631.  The  Appraiser  examines  imported  articles,  and  determines  their  duti- 
able value  and  also  the  rate  of  duty  to  be  charged. 

632.  The  Store-keeper  has  charge  of  the  warehouse. 

Remarks. — 1.  Warehousing  is  depositing  imported  goods  in  a  government  or  bonded  ware- 
house. 

2.  A  bonded  warehouse  is  used  for  storing  goods  on  which  the  duties  have  not  been  paid. 

3.  Goods  may  be  withdrawn  from  a  bonded  warehouse  for  export,  without  the  payment  of 
the  duties.  If  goods  on  which  the  duty  has  been  paid  are  exported,  the  amount  of  duty  so 
paid  is  refunded;  the  sum  so  refunded  is  called  a  dratclacJ:. 

4.  Smu^jQling  is  bringing  foreign  goods  into  the  country  without  paying  the  required  duty. 
This  is  done  either  by  not  entering  them  at  a  Custom-House,  or  by  showing  less  than  their  real 
value  in  the  invoice.  It  is  a  crime,  for  the  prosecution  and  punishment  of  which  stringent 
laws  are  enacted. 

5.  Many  merchants  employ  a  Custom-House  Broker,  one  familiar  with  the  laws,  to  enter 
goods  for  them . 

633.  To  find  Specific  Duty. 

Example. — What  is  the  specific  duty  on  1-40  casks  of  alcohol,  of  60  gallons 
each,  at  15'/  per  gallon;  leakage  5^? 

Operatiok. 
140  X  60  gal.  =  S400  gal. 

05  =  'fc  of  leakage  Explakatiox. —  Specific  duty  is  computed 

— —                            °  '  on  the  net  quantity;  to  find  the  net  quantity, 

420  gal.  =  leakage.  t^ke  420  gallons,  the  allowance  for  leakage, 

8400  gal.  =  ^ross  quantitv.  ^^"""^  *^-  ^^^  ^^^^le  number  of  gallons,  which 

.    r    I-                   '  gives  7980  gallons,  on  which  to  charge  duty. 

^"^'  ^''^  '  ~  lea^^^o^'  .Since  the  duty  is  15  cents  per  gallon,  for  7980 

7980  =  net  quantity.  gallons  it  wiU  be  $1197. 
S.15  =  duty  per  gallon. 


81197.00  =  specific  duty. 


EXAMPLES  ISr   CUSTOM-HOUSE   BUSINESS.  199 

Rule. — Multiply  the  net  quantity  Inj  the  duty  per  single  article  of  the 
Jcind  or  class  considered. 

EXAMPLES   FOK   PKACTICE. 

634.  1.  AVhat  is  the  duty  on  CO  packages  of  figs,  each  of  16  lb.  weight,  at 
2-^^'  per  pound,  tare  5^? 

2.  Find  the  duty,  at  75^;  per  ton,  on  an  invoice  of  897130  lb.  of  bituminous 
«oal. 

3.  If  tlie  duty  on  phite  glass  is  25^'  per  square  foot,  how  much  will  be  charged 
on  an  importation  of  200  boxes,  each  containing  20  plates  24  X  48  in.  in  size  ? 

4.  Find  the  duty,  at  12  per  dozen,  on  40  doz.  bottles  of  wine  imported  from 
Lyons,  if  an  allowance  of  10,'^  is  made  for  breakage. 

5.  If  the  duty  is  65^  per  cubic  foot,  what  amount  must  be  paid  on  an  impor- 
tation of  G  blocks  of  marble,  each  10  ft.  long,  3  ft.  wide,  2  ft.  high  ? 

6.  After  being  allowed  10,^  for  leakage,  a  wine  merchant  paid  $864  duty,  at 
t2  per  gallon,  on  12  casks  of  wine.  How  many  gallons  did  eacli  cask  originally 
contain  ? 

7.  Find  the  duty  on  1500  dozen  empty  bottles,  breakage  4^,  and  rate  of  duty 
10^  per  dozen. 

635.  Applying  the  terms  of  Percentage  to  Ad  Valorem  Duties,  we  observe 
the  following  : 

The  net  Value,  or  Quantity  =  the  Base. 
The  Rate  Per  Cent.  Ad  Valorem  =  the  Rate. 
Tlie  Duty  =  the  Percentage. 

636.  To  find  Ad  Valorem  Duty. 

Example. — What  is  the  ad  valorem  duty,  at  35^,  on  90  boxes  of  brass  rivets, 
^6  lb.  per  box,  invoiced  at  12^'  per  pound,  tare  being  6  lb.  j)er  box  ? 

Operation. 

90  X  25  lb.  =  2250  lb.  gross  weight. 
90  X     6  lb.  =    540  1b.  tare. 

1710  lb.  net  weight. 

1710  Explanation,— Find  the  net  weight  and 

$.12  ^=  cost  per  pound  value  as  the  base;    multiply  by  the  rate  of 

$205.20  =  net  value.  ^''*^- 

.35        =  ^  of  duty. 
1025 
615 


$71.75  =  duty. 

Remark. — The  cents  in  the  net  value,  being  less  than  50,  are  rejected. 

Rule. — Multiply  the  vfdue,  after  all  deductions  are  made,  by  the  per 
cent,  of  duty  assessed. 


200  EXAMPLES   IN"   CUSTOM-HOL'SE   BUSINESS. 

EXAMPLES  FOR  PRACTICE. 

637.  1.  What  is  the  duty,  at  50j^,  upon  a  consignment  ot  650  dozen  kid 
gloves,  invoiced  at  90  francs  per  dozen  ? 

2.  An  importation  of  English  crockery  was  invoiced  at  £896,  5s.  6d.  Find 
the  duty,  at  40^. 

3.  If  an  importation  is  invoiced  at  174-50  francs,  what  will  be  the  duty,  at  35^?' 
^.     If  the  duty  on  sperm  oil  is  20^,  what  will  it  amount  to  in  an  importation 

of  600  barrels,  of  42  gallons  each,  invoiced  at  450  per  gallon,  3|^  being  allowed 
for  leakage  ? 

5.  I  received  by  steamer  Raglan,  from  Liverpool,  the  following  invoice  of 
goods  :  7G8  yd.  velvet,  invoiced  at  £1  12s.  per  yd. ;  2150  yd.  lace,  invoiced  at 
3s.  4d.  per  yd.;  1200  yd.  broadcloth,  invoiced  at  15s.  per  yd. :  3520  yd.  carpet, 
invoiced  at  lis.  Gd.  per  yd.  Jf  the  duty  on  the  velvet  was  eO'v',  on  the  lace  and 
broadcloth  35^,  and  on  the  carpet  50^,  how  much  was  the  total  duty  to  be  paid?" 

MISCKLtANEOrS  EXAMPLES. 

638.  1.  What  is  the  duty  on  1000  yd.  of  brussels  carpet,  27  in.  wide,  invoiced 
at  6s.  9d.  per  yd;  duty  440  per  square  yard  specific,  and  35,'^  ad  valorem  ? 

2.  If  the  duty  on  flannel  is  290  per  pound  specific,  and  35^  ad  valorem,  how 
much  must  be  paid  on  an  invoice  of  2150  yd.,  weighing  •i20  lb.,  and  valued  in 
Canada,  whence  it  was  imported,  at  750  per  yard  ? 

3.  Find  the  duty  on  3  dozen  clocks,  invoiced  at  $21.50  eacli,  and  6  dozen 
watches,  invoiced  at  135  each,  if  the  ad  valorem  duty  was  35^"^  on  the  clocks,  and 
25f^  on  the  watches. 

Jf..  How  much  duty  must  be  paid  on  an  importation  of  27640  lb.  of  wool, 
invoiced  at  £1497  10s.  4d.,  if  the  rate  of  duty  is  100  per  pound  specific,  and  11^ 
ad  valorem? 

5.  I  imported  from  Canada  7240  bushels  of  barley,  and  17^  tons  of  hay, 
invoiced  at  |i9.50  per  ton.  What  amount  of  duties  had  I  to  pay,  at  100  per 
bushel  on  the  barley  and  20,^^  on  the  hay? 

6.  A  merchant  imported  300  pieces  of  three-ply  carpet,  each  piece  containing 
75  sq.  vd.,  invoiced  at  3s.  6d.  per  square  yard,  upon  which  he  paid  a  duty  of 
170  per  square  yard  specific,  and  35,'*  ad  valorem.  What  was  the  total  amount 
of  duty  paid  ? 

7.  An  invoice  of  woolen  cloth,  imported  from  England,  was  valued  at  £956 
6s.  If  its  weight  was  684  lb.,  how  much  was  the  duty,  at  500  per  pound  specific, 
and  35,'fc  ad  valorem  ? 

8.  Find  the  duty  on  50  cases  of  tobacco,  each  weighing  60  lb,,  and  50000 
Havana  cigars  weighing  550  lb.,  invoiced  at  175  per  M,  the  duty  being  500  per 
pound  specific  on  the  tobacco,  and  $2.50  per  pound  specific  and  25f^  ad  valorem 
on  the  cigars. 


TAXES.  201 


TAXES. 

639.  Taxes  are  sums  of  money  levied  on  persons,  property,  or  products,  for 
any  public  purpose. 

640.  Capitation  or  Poll  Taxes  are  levied  at  a  certain  amount  for  each 
person  or  head  of  legal  voters  not  exempt  by  law. 

641.  Property  Tax  is  a  tax  assessed  or  levied  upon  property,  at  a  giveii  rate 
per  cent,  of  the  valuation. 

642.  Property  is  of  two  kinds:  Personal  and  Real. 

643.  Personal    Property  is  movable  property ;    as,   merchandise,   ships, 

cattle,  money,  stocks,  mortgages,  etc. 

644.  Real  Property  or  Real  Estate  consists  of  immovable  property;  as, 

houses  and  lands. 

645.  Assessors  are  public  or  government  officers,  who  appraise  the  value  of 
property  to  be  taxed,  and  apportion  the  ta«es  pro  rata;  that  is,  in  proportion  to 
the  value  of  each  man's  property. 

646.  Collectors  are  i)ublic  or  government  officers,  who  collect  taxes. 

Remark.— Taxes  are  generally  assessed  and  made  payable  in  money,  but  in  "road  taxes" 
they  may  be  made  payable  in  "day's  work." 

647.  The  terms  of  Percentage,  applied  to  Taxes,  are: 
The  Valuation  =  the  Base. 

The  tax  on  $1.00  =  the  Rate. 

The  Sum  to  be  raised  =  the  Percentage. 

The  Sum,  minus  the  Collector's  fees,  or  commission  =  the  Difference.     ' 

648.  To  find  a  Property  Tax. 

Example.  — The  rate  of  taxation  in  the  city  of  Des  Moines,  Iowa,  is  If^. 

What  amount  of  tax  must  a  person  pay,  whose  personal  property  is  valued  at 

$17500,  and  who  owns  real  estate  assessed  at  $24:900  ? 

Operation. 

$17500  Explanation.— Since  his  total  valuation  was  ^42400, 

c)Aar\r\  ''"^  ^^^'  ^^^^  "^  taxation  1|  per  cent.,  his  tax  would  be  If 

per  cent  of  $42400,  or  $742. 
$42400  X  .011  =  $742. 

Rule. — Multiply  the  total  assessed  value  by  the  rate  per  cent,  of  taxation. 

EXAMPLES   FOK   PRACTICE. 

649.  1.  Mr.  R.  owns  personal  property  assessed  at  $7140,  and  real  estate 
assessed  at  $11100,  in  a  village  in  which  he  is  taxed  one-half  of  one  per  cent. 
Find  the  amount  of  his  tax. 


202  EXAMPLES   IX   TAXES. 

2.  A  person  having  lands  valued  at  $7500,  #12*250  in  money,  and  a  stock  of 
goods  worth  f6000,  pays  tax  on  all  at  the  rate  of  U<.     Find  his  total  tax. 

650.  To  find  a  General  Tax. 

Example. — A  tax  of  f!250o  is  to  be  assessed  upon  the  village  of  Livonia  ;  the 
valuation  of  the  taxable  property  is  $000000,  and  there  are  S'U  polls,  to  be  tissessed 
SI.  25  each.  What  will  be  the  tax  on  a  dollar,  and  how  much  will  be  the  tax  of 
Mr.  Scott,  whose  property  is  valued  at  $12500,  and  who  pays  for  2  polls. 

Operation. 
$1.25  X  324  =  S405,  amount  of  poll  tax. 
$2505  —  $405  =  $2100,  amount  of  property  tax. 
$2100  -^  $600000  =z  .0035,  rate  of  taxation. 
$12500  X  .0035  =  $43.75,  Mr.  Scott's  property  tax. 
$43.75  -f  $2.50  (2  polls)  =  $46.25,  Mr.  Scott's  total  tax. 

Expla>"atiox. — Since  $2505,  the  amount  to  be  raised,  includes  both  the  poll  and  property 
tax,  if  $405,  the  poll  tax,  is  subtracted  from  this  amount,  the  remainder,  $2100,  will  be  the 
Percentage,  or  sum  to  be  assessed  on  the  Base,  or  entire  property.  Divide  this  Percentage  by 
this  Base,  and  the  quotient  will  be  the  rate  of  tax  assessed,  3i  mills  on  the  dollar.  Multiply 
$12500,  the  assessed  valuation  of  Mr.  Scott's  property,  by  .0035,  the  per  cent,  expressed  deci- 
mally, and  the  result,  $43.75,  is  his  property  tax;  adding  to  this  $2.50,  the  tax  on  two  polls, 
gives  $46.25,  his  entire  tax. 

Rule. — Fvom  the  sum  to  be  raised,  deduct  the  -poll  tax,  if  any ;  divide 
the  remainder  by  the  total  assessineiit,  and  multiply  the  assessment  of 
each,  individual  by  the  quotient;  add  to  the  product  the  aitvount  of  pdlZ 
tax  to  be  paid. 

EXAMPLES  FOR   PRACTICE. 

651.  1.  A  tax  of  $125000  is  levied  on  a  city,  the  assessed  valuation  of  which 
is  $15000000.  "What  is  the  rate  of  taxation,  and  what  amount  of  tax  will  a 
t>erson  have  to  pay  whose  property  is  valued  at  $7500  ? 

2.  If  a  tax  of  $120  is  assessed  on  a  mill  valued  at  $24000,  what  is  the  valua- 
tion of  a  residence  that  is  taxed  $17.75  at  the  same  rate  ? 

3.  The  per  cent,  of  tax  assessed  for  state  purposes  is  i^,  for  county  ^^,  and  for 
citv  1|^.     What  will  be  the  amount  of  my  tax,  on  property  assessed  at  $21500  ? 

4.  The  tax  assessed  upon  a  town  is  $20914.80;  the  town  contains  2580  polls, 
taxed  $.624  each,  and  has  a  real  estate  valuation  of  $4062000,  and  a  valuation 
of  i)ersonal  property  to  the  amount  of  $227400.  Find  the  rate  of  taxation,  and 
C's  tax,  who  pays  for  4  polls,  and  whose  property  is  assessed  at  $15000. 

Remark.— In  certain  States,  the  common  schools  are  supported  by  a  tax  or  rate  bill  made 
out  en  the  basis  of  the  total  attendance. 

o.  My  son  and  daughter  each  attended  school  214  days,  and  the  expense, 
including  teachers  wages  and  incidentals,  was  paid  by  a  rate  bill.  How  much 
must  I  pay,  if  the  teacher's  wages  amounted  to  $440,  fuel  and  repairs  $101.50, 
and  janitors  fees  $74.75,  and  the  total  number  of  day's  attendance  was  7460? 


EXAMPLES   IN   TAXES.  203 

6.  For  the  year  1888  the  rates  of  taxation  in  the  State  of  New  York  were 
as  follows:  Schools,  1.085  mills;  general  purposes,  1.475  mills;  new  capitol,  .G 
of  a  mill;  other  purposes,  .34  of  a  mill.  What  was  the  total  rate  of  taxation, 
and  hoAv  much  was  raised  by  Livingston  County,  the  valuation  of  which,  as  fixed 
i)y  the  State  board  of  equalization,  was  $25395180?  How  much  did  said  county 
raise  for  school  purposes? 

7.  The  cost  of  maintaining  the  public  schools  of  a  city  during  the  year  1888, 
was  $112000,  and  the  taxable  property  of  the  city  was  $44800000.  How  many 
mills  on  a  dollar  must  be  assessed  for  school  purposes?  If  10^  of  the  tax  assessed 
cannot  be  collected,  liow  mauy  mills  on  a  dollar  must  then  be  assessed  ? 

8.  A  tax  of  $13943.20  is  assessed  upon  a  town  containing  8G0  taxable  i)olls; 
the  real  estate  is  valued  at  $2708000,  and  the  personal  property  at  $151000.  If 
the  polls  be  taxed  $1.25  each,  what  will  bo  the  rate  of  property  taxation,  and 
what  will  be  the  tax  of  Peter  Parley,  Avho  pays  for  three  i)olls,  and  has  real  and 
personal  estate  valued  at  $23750? 

9.  In  a  school  district,  the  valuation  of  the  taxable  property  is  $752400,  and 
it  is  proposed  to  repair  the  school  house  and  ornament  the  grounds,  at  an  expense 
of  $5000.  If  old  material  sells  for  $073.70,  what  will  be  the  rate  per  cent,  of 
taxation,  and  what  will  be  B's  tax,  whose  i)roperty  was  valued  at  $9400? 

10.  The  assessed  value  of  a  town  is,  on  real  estate,  $1197500,  and  on  personal 
property,  $432500.  A  poll  tax  of  $.50  per  head  is  assessed  on  each  of  1870 
persons.  The  town  votes  to  raise  $8000  for  schools,  $1500  for  highways,  $1500 
for  salaries,  $1000  for  support  of  poor,  and  $310  for  contingent  expenses.  How 
much  tax  Avill  a  milling  company  have  to  pay,  on  a  mill  valued  at  $46500,  and 
stock  at  $19750? 

11.  The  total  assessed  value  of  a  town,  real  and  personal,  is  ^630000,  and  the 
town  expenses  are  $3913.95.  How  much  tax  must  be  collected  to  provide  for 
town  expenses  and  allow  3^  for  collecting?  If  the  same  town  contains  310  polls, 
taxed  $1.50  each,  what  will  be  the  rate  of  taxation,  and  how  much  will  be  the 
tax  of  a  man  who  pays  for  two  polls  and  owns  property  assessed  at  $14500  ? 

13.  The  assessed  valuation  of  the  real  estate  of  a  county  is  $1910887,  of  the 
personal  property,  $921073,  and  it  has  4564  inhabitants  subject  to  a  poll  tax. 
The  years  expenses  arc:  for  schools,  $8400;  interest,  $6850;  highways,  $7560; 
salaries,  $5150;  and  contingent  expenses,  $13675.  If  the  poll  tax  was  $1.50,  and 
the  revenue  from  fairs  and  licenses  $6200,  what  tax  must  be  levied  on  a  dollar  to 
meet  expenses  and  provide  a  sinking  fund  of  $7000? 


204  lilSUEANCE. 


INSURANCE. 

652.  Insurance  is  indemnity  secured  against  loss  or  damage.  It  is  of  two 
kinds:   Property  Insurance  and  Personal  Insurance. 

653.  Property  Insurance  includes: 

1.  Fire  Insurance,  or  indemnity  for  loss  of  or  damage  to  property  by  fire. 
~.     Marine  Insurance,  or  indemnity  for  loss  of  or  damage  to  a  ship  or  its 

cargo,  by  any  specified  casualty,  at  sea  or  on  inland  waters. 

3.  Live  Stock  Insurance,  or  indemnity  for  loss  of  or  damage  to  horses, 
cattle,  etc.,  from  lightning  or  other  casualty. 

654.  The  Insured  Party  is  usually  the  owner  of  the  property  insured,  but 
may  be  any  person  having  a  financial  insurable  interest  in  the  property. 

655.  The  Insuring  Parties  are  called  Insurers  or  Underwriters,  and  are 
usually  incorporated  companies. 

656.  Insurance  Companies  are  distinguished  by  the  way  in  which  they 
are  organized;  as  Stoch  Insurance  Companies,  Mutual  Insurance  Companies. 

657.  A  Stock  Insurance  Company  is  one  whose  capital  has  been  con- 
tributed and  is  owned  by  stockholders,  who  share  the  profits  and  are  liable  for 
the  losses. 

658.  A  Mutual  Insurance  Company  is  one  in  which  the  profits  and  losses 
are  shared  by  the  insured  parties. 

Remarks. — 1.  Some  companies  combine  the  features  of  both  stock  and  mutual  companies, 
and  are  called  Mixed  Companies. 

2.  In  mixed  companies,  all  profits  above  a  limited  dividend  to  the  stockholders  are  divided 
among  the  policy-holders. 

659.  Transit  Insurance  refers  to  risks  taken  on  goods  being  transported 
from  place  to  place,  cither  by  rail  or  water  or  both. 

660.  The  Policy  is  the  contract  between  the  insurance  company  and  the 
person  whose  proi)erty  is  insured,  and  contains  a  description  of  tbe  insured 
property,  the  amount  of  the  insurance,  and  the  conditions  under  which  the  risk 
is  taken, 

661.  The  Premium  is  the  consideration  in  the  contract,  or  the  sum  i)aid 
for  insurance. 

662.  The  Term  of  Insurance  is  tlie  period  of  time  for  which  the  risk  is 
taken,  or  the  property  insured. 

Remarks. — 1.  Premium  rates  are  usually  given  as  so  much  per  $100  of  the  sum  insured, 
and  depend  upon  the  nature  of  the  risk  and  the  length  of  time  for  which  the  policy  is  issued; 
insurance  is  usually  effected  for  a  year  or  a  term  of  years. 

2.  Short  Rates  are  for  terms  less  than  one  year. 

3.  It  is  usual  to  make  an  added  charge  for  the  policy. 

4.  Insurance  is  frequently  effected  upon  plate  glass,  the  acts  of  employees,  etc. 


INSURANCE.  205 

663.  An  Insurance  Agent  is  one  who  acts  for  an  insurance  company,  in 
obtaining  insurance,  collecting  premiums,  adjusting  losses,  reinsuring,  etc. 

664.  An  Insurance  Broker  is  a  person  who  negotiates  insurance  for  others, 
for  wliich  he  receives  a  brokerage  from  the  company  taking  the  risk;  he  is  con- 
sidered, however,  an  agent  of  the  insured,  not  of  the  company. 

Remark. — A  Floating  Policy  is  one  which  covers  goods  stored  in  different  places,  and  gen- 
erally such  as  are  moved  from  place  to  place  in  process  of  manufacture. 

665.  Losses  may  be  total  or  imrtial. 

666.  Fire  Insurance  Losses  are  usually  adjusted  by  the  insurance  company 
paying  the  full  amount  of  the  loss,  provided  such  loss  does  not  exceed  the  sum 
insured;  if  the  policy,  however,  contains  the  "average  clause,"  the  payment 
made  is  such  proportion  of  the  loss  as  the  amount  of  insurance  bears  to  tlie  total 
value  of  the  property. 

667.  When  a  loss  occurs  to  a  vessel,  the  insurance  company  pays  only  such 
a  proportion  of  the  loss  as  the  policy  is  of  the  entire  value  of  the  vessel. 

668.  It  is  an  established  rule  in  marine  insurance,  that  insurers  shall  be 
allowed  one-third  for  the  superior  value  of  the  new  material,  as  sails,  masts,  etc., 
used  in  repair  of  damage;  that  is,  "one-third  off  new  for  old." 

Remark. — Marine  policies  usually  contain  the  "average  clause." 

669.  In  case  a  policy  is  terminated  at  the  request  of  the  insured,  he  is  charged 
the  "short  rate  "  premium;  if,  however,  it  be  terminated  at  the  option  of  the 
company,  the  lower  long  rate  will  be  charged,  and  the  compan)'  refund  the 
premium  for  the  unexpired  time  of  the  policy. 

670.  A  Talued  or  Closed  Policy  is  the  ordinary  form,  and  contains  a  tixed 
valuation  of  the  thing  insured. 

671.  An  Open  Policy  is  one  upon  which  additional  insurances  may  be 
entered  at  any  time  from  port  to  port,  at  rates  and  under  conditions  agreed  upon. 

672.  Policies  on  Cargoes  are  issued  for  a  certain  voyage,  and  on  vessels 
for  a  voyage  or  for  a  specified  time.     • 

673.  Salvage  is  an  allowance  made  to  those  rendering  voluntarv  aid  in 
saving  vessels  or  cargoes  from  marine  casualties. 

Remarks. — 1.  Insurance  companies  usually  reserve  the  privilege  of  rebuilding,  replacing, 
or  repairing  damaged  property. 

2.  Insurance  policies  ordinarily  state  that  the  loss,  if  becoming  a  charge  upon  the  c-ompany, 
-will  be  paid  30  days  or  GO  days  after  due  notice  and  proof  of  loss.  If  not  then  paid,  the  amount 
of  the  claim  becomes  Interest-bearing. 

674.  The  computations  in  Property  Insurance  are  performed  the  same  as  in 
Percentage,  and  the  terms  compare  as  follows: 

The  Amount  Insured  =  the  Base. 
TheRate  ^/o  of  Premium  =  the  Rate. 
The  Premium  =  the  Percentage. 


206  EXAMPLES    IN    INSURANCE. 

675.     To  find  the  Cost  of  Insurance. 

Example. — The  mixed  stock  in  a  country  store  is  insured  for  $7500.  What 
is  the  cost  of  insurance  for  one  year,  at  1^^  premium,  if  $1.25  is  charged  for 
the  policy? 


Opkration. 
17500.  =  amount  insured. 


Explanation. — Since  the  amount  insured  is 

the  base,  and  the  per  cent,  of  premium  the  rate, 

•Q^'^     =  r^  of  premium.  jf  ^l^^,  amount  be  multiplied   by  the   rate,   the 

$11.25    =  premium.  product,   $11.25,  will  be  the  premium;  adding 

1.25    =  cost  of  ijolicy.  $1.25,  the  cost  of  the  policy,  gives  the  full  cost, 

'' $12.50 

$12.50    =  full  cost  of  insurance. 

Jiule.— Multiply  the  amount  of  insurance  hy  the  rate  per  cent,  of 
premium,  anil  add  extra  charges,  if  any. 

676.  To  find  the  Amount  Insured,  the  Premium  and  Per  Cent,  of  Premium 
being  given. 

Example. — I  paid  $141.50  to  insure  a  stock  of  goods  for  three  months.  If 
the  charge  for  the  policy  was  $1.50,  and  the  rate  of  premium  ^^,  for  what  amount 
was  the  policy  issued? 

Operation. 

Explanation.— Since  $141.50  was  the  full 

$141.50  =  full  cost.  (,Qgt  oj.  premium  plus  the  charge  of  $1.50  for 

1-50  =  cost  of  policy.  the  policy,  the  premium  must  have  been  $140; 

$140        ;=  premium.  ^°*i  since  the  rate  of  premium  was  |  per  cent., 

l^  =  .00875  =  decimal  rate.  '^  ^^^^  '•'  ^^^^^^^  ^^  ^  P^""  ^^°'-  *^^  Quotient, 

A-i  Ar^         t\r\c,^~        a.i  r>r\,^n  -p  j?       T  $16000,  will  be  the  facc  of  the  policy. 

$140 -=- .0087o  =  $16000,  face  of  policy.    ^        '  ^      ^ 

Rule.  From,  the  full  cost  of  insurance,  subtract  the  extra  charges,  if 
any;  divide  the  remainder  hy  the  per  cent,  of  premium,  and  the  quotient 
will  he  the  face,  of  the  policy. 

EXAMPLES   rOR    I'KACTICE. 

677.  J.     How  much  insurance,  at  \\^,  can  be  procured  for  $62.50? 

2.  A  ranchman  paid  a  premium  of  $75.20  for  insuring  f  of  his  herd  of 
cattle,  at  60^-  per  $100.  If  the  cattle  were  valued  at  $40  per  head,  how  many 
had  he? 

3.  The  loss  on  a  property  was  $6000,  of  which  $2000  was  insured  in  the 
Home,  $3000  in  the  Phmnix,  and  $2500  in  the  Hartford.  How  much  did  each 
company  contribute? 

Jf.  If  it  cost  $663  to  insure  a  certain  block  for  $44200,  what  will  be  the 
cost,  at  the  same  rate,  to  insure  a  block  valued  at  $105000,  if  $1.50  extra  be 
charged  for  the  policy  in  the  latter  case? 

J.  How  much  will  it  cost  to  insure  a  factory  for  $42000,  at  f^r,  and  its 
machinery  for  816500,  at  \\'/<,y  charge  for  policy  and  survey  being  $2.50? 

6.  A  gentleman  paid  835.60  per  annum  for  insuring  his  house,  at  2f^  on  two 
fifths  of  its  value.     What  was  the  value  of  the  house? 


EXAMPLES    IX    INSURANCE.  207 

7.  If  a  store  and  its  contents  are  valued  at  $27000,  for  how  much  must  it  be 
insured,  at  H^  to  cover  loss  and  premium  in  case  of  total  destruction? 

8.  A  cargo  of  teas,  valued  at  8330^)0,  was  insured  for  $18000,  in  a  policy 
containing  an  "average  clause."  In  case  of  damage  to  tlie  amount  of  $21000, 
how  much  should  the  company  pay? 

9.  The  steamer  Norseman,  valued  at  $90000,  is  insured  for  $75000,  at  2^^. 
What  will  be  the  actual  loss  to  the  insurance  company,  in  case  the  steamer  is 
damaged  to  the  amount  of  $20000? 

10.  A  speculator  bought  2000  barrels  of  flour,  and  had  it  insured  for  80<^.  of 
its  cost,  at  34^,  paying  a  premium  of  $429.  At  what  price  must  he  sell  the  flour, 
to  make  a  net  profit  of  10«^? 

11.  I  insured  my  grocery  store,  valued  at  $13500,  and  its  contents,  valued  at 
$33000,  and  paid  $350  for  premium  and  policy.  If  the  policy  cost  $1.25,  what 
was  the  rate  per  cent,  of  premium? 

12.  A  canal-boat  load  of  8400  bushels  of  wheat,  worth  90^'  i)er  bushel,  is 
insured  for  three-fourths  of  its  value,  at  If^  premium.  In  case  of  the  total 
destruction  of  the  wheat,  how  much  will  the  owner  lose  ? 

13.  A  stock  of  goods,  valued  at  $30000,  was  insured  for  18  months,  at  1\'^;  at 
the  end  of  12  montlis  tlic  owner  surrendered  the  policy.  If  the  "short  rate" 
for  6  months  was  65^  per  $100,  what  should  be  the  return  jiremium? 

H.  For  how  much  must  a  house  worth  $G000,  and  furniture  worth  $2000,  be 
insured,  at  1^  per  cent.,  to  cover  the  cost  of  the  policy,  which  was  $2,  the 
amount  of  premium  paid,  and  f  of  the  value  of  the  property? 

15.  A  man  owning  |  of  a  ship,  insured  f  of  his  interest,  at  l\fc,  and  i)aid 
$91.50  for  premium  and  a  policy  charge  of  $1.50.  If  the  ship  becomes  damaged 
to  the  extent  of  $12000,  how  much  can  be  recovered  on  the  policy? 

16.  A  schooner  is  valued  at  $10500,  and  has  a  cargo  of  3500  barrels  of  apples, 
worth  $2.10  per  barrel.  What  amount  of  insurance  must  be  obtained,  at  'i^ii, 
to  provide,  in  case  of  loss,  for  the  value  of  the  property,  the  premium,  and  $5 
additional  which  the  owner  paid  for  survey  and  policy? 

17.  A  block  of  stores  and  contents  was  insured  for  $220000,  and  became  dam- 
aged by  fire  and  water  to  the  amount  of  $150000.  Of  the  risk,  $40000  was  taken 
by  the  Hartford  Co.,  $05000  by  the  Manhattan,  $35000  by  the  yEtna,  and  the 
remainder  was  divided  equally  between  the  Piuenix  and  the  Provident.  What 
was  the  net  loss  of  each  company,  if  the  premium  paid  was  1^;^? 

18.  The  furniture  in  my  house  is  estimated  at  one-half  the  value  of  the  house. 
I  get  both  insured  for  $7687.50  for  5  years,  at  24f?;,  and  find  that  in  case  of  total 
destruction  the  face  of  the  policy  will  be  full  indemnity  for  both  the  property 
and  premium.     Find  the  value  of  the  house. 

19.  A  factory  worth  $45000  is  insured,  with  its  contents,  for  $62500;  $30000 
of  the  insurance  is  on  the  building.  $12500  on  machinery  worth  $20000,  and 
$20000  on  stock  worth  $35000.  A  fire  occurs  by  Avhich  the  building  and  tlie 
machinery  are  both  damaged,  each  to  the  amount  of  $15000,  and  the  stock  is 
entirely  destroyed.  How  much  is  the  claim  against  the  company,  if  the  risk  is 
covered  by  an  "ordinary"  policy?  How  much  if  tlie  i)olicy  contains  the  "aver- 
age clause?" 


308  PERSONAL   INSURANCE. 

20.  The  German  Insurance  Company  insured  the  Field  block  for  $105000,  at 
60^  per  $100;  but  thinking  the  risk  too  great,  it  re-insured  $40,000  in  the  Home, 
at  f*?^,  and  $45000  more  in  the  Mutual,  at  ^'i.  How  much  premium  did  each 
company  receive?  What  -svas  the  gain  or  loss  of  tiie  German?  "Wlnit  per  cent, 
of  premium  did  it  receive  for  the  part  of  the  risk  not  re-insured? 


PERSONAL    INSURANCE. 

678.  Personal  Insurance  is  tlie  insurance  of  ])ersons.     It  includes: 

1.  Life  Insurance,  or  indemnity  for  loss  of  life. 

2.  Accident  Insurance,  or  indemnity  for  loss  from  disability  occasioned 
by  accident. 

3.  Jlealth  Insurance,  or  indemnity  for  loss  occasioned  by  sickness. 

679.  Policies  of  Life  Insurance  are  usually  either  Life  Policies  or 
Endoicmeut  Policies. 

680.  A  Life  Policy  stipulates  to  pay  to  the  beneficiaries  named  in  it  a  fixed 
sum  of  money  on  the  death  of  the  insured. 

681.  An  Endowment  Policy  guarantees  the  payment  of  a  fixed  sum  of 
money  at  a  specified  time,  or  at  death,  if  the  death  occurs  before  the  specified 
time. 

682.  Life  insurance  companies  are  known  as  Stock,  Mutual,  Mixed,  and 
Co- Operative. 

683.  Losses  sustained  by  Stock  and  Mixed  companies  are  jiaid  either  from 
*' reserve  funds"  or  by  assessment  on  the  stockholders;  those  sustained  by 
Mutual  and  Co-Operative  companies  are  paid  by  pro-rata  or  fixed  contributions 
of  the  policy  holders. 

Remarks. — 1.  The  money  may  be  made  payable  to  any  one  named  by  the  insured;  if  made 
payable  to  himself,  at  his  death  it  becomes  a  part  of  his  estate  and  is  liable  for  his  debts,  if 
payable  to  another,  that  other  cannot  be  deprived  of  the  benefit  of  the  insurance,  either  by  the 
will  of  the  person  taking  out  the  insurance,  or  by  his  creditors. 

2.  A  person  may  insure  his  life  in  as  many  companies  as  he  pleases,  and  to  any  amount. 

3.  Anj'  one  having  an  insurable  interest  in  the  life  of  another,  may  take  out,  hold,  and  be 
benefited  by  a  policy  of  insurance  upon  the  life  of  the  other;  or  he  may  take  out  a  policy  in 
his  own  name,  and  then  assign  it  to  any  creditor  or  to  anj-  one  having  an  insurable  interest. 

4.  The  practical  workings  of  life  insurance  are  fully  set  forth  in  documents  in  general  circu- 
lation, and  all  matters  of  premiums  to  be  paid,  cash  value  of  policies  surrendered,  and  manner 
of  becoming  insured,  are  determined  from  such  documents,  rendering  it  unnecessary  to  require 
the  solution  of  problems  under  life  insurance, 


INTEREST.  209 


INTEREST. 

•684:.     Interest  is  a  compensation  paid  for  the  use  of  money. 

685.  The  Principal  is  tlie  money  for  the  use  of  whieli  interest  is  paid. 

686.  The  Anionnt  is  the  sum  t)f  the  princijMil  and  interest. 

687.  The  Time  is  the  jjeriod  during  which  the  principal  bears  interest. 

688.  Interest  is  reckoned  at  a  certain  per  cent,  of  tlie  principal.  It  is 
therefore  a  Per  Cent,  of  Avhich  the  Base  is  the  Principal. 

689.  The  Rate  of  Interest  is  the  annual  rate  per  cent. 

690.  Interest  differs  from  the  preceding  applications  of  Percentage  only 
.by  introducing  time  as  an  element,  in  connection  with  the  rate  per  cent. 

The  Principal  =  the  Base. 

The  Per  Cent,  per  Annum  =  the  Rate. 

The  Interest  =  tlie  Percentage. 

The  Sum  of  the  Principal  and  Intei;est  =  the  Amount. 

691 .  Legal  Interest  is  interest  according  to  the  maximum  rate  fixed  by 
law. 

692.  Tsury  is  interest  taken  at  a  rate  liigher  than  the  law  allows. 

693.  Simple  Interest  is  interest  on  tlie  priiu-ipal  only,  for  the  whole  time 
•of  the  loan  or  credit;  and  this  is  generally  understood  by  the  term  interest. 

694.  Annual,  Semi -Annual,  or  other  Periodic  Interest,  is  interest 
•computed  at  a  specified  rate  for  a  year,  half-year  or  other  designated  period. 

69.5.  (*ompoun(l  Interest  is  interest  computed  on  the  amount  at  ri>gular 
intervals. 

Remabks. — 1.  The  payment  of  periodic  interest,  if  specified  in  a  contract,  may  usually  be 
enforced;  and  if  not  paid  when  due,  becomes  simple  interest  bearing,  and  is  not  usury. 

2.  Neither  the  paying  nor  the  receiving  of  compound  interest  is  usury;  but  its  payment 
cannot  ordinarily  be  enforced,  even  though  it  is  mentioned  in  the  contract. 

696.  Accrued  Interest  is  interest  accumulated  on  account  of  any  obliga- 
tion, due  or  not  due. 

69.7.  Conimoii  Interest  is  interest  comi)uted  on  a  basis  of  360  days  for  a 
year. 

Remarks. — 1.  This  method  is  generally  employed  by  business  men,  and  in  some  states  has 
received  the  sanction  of  law. 

2.  In  reckoning  interest  l)y  this  method,  it  is  customary  to  consider  a  year  to  be  12  months, 
and  a  month  to  be  30  days. 

Statement.— July  22,  1887.  at  the  annual  convention  of  the  Business  Educators'  Associa- 
tion of  America,  then  in  session  at  Milwaukee,  Wis.,  the  following  resolution  was  unanimously 
14 


2]0  SIX    PER   CENT.    METHOD. 

adopted  :  Rewired,  That,  as  business  educators,  we  uniformly  teach  interest  and  discount  on 
a  360-day  basis,  finding  time  by  compound  subtraction,  and  calling  each  month  thirty  days, 
except  where  the  day  of  the  minuend  time  be  thirty-on^,  when  it  shall  be  so  counted. 

RiiMABK. — In  computing  interest  for  short  periods  of  lime,  it  is  customary  to  take  the  exact 
numl)er  of  days. 

698.  Exact  luterest  is  iuterest  computed  for  the  exact  time  in  days,  and 
regarding  the  days  as  3G5ths  of  a  year.  This  method  is  used  by  the  United 
States  Government  and  by  some  merchants  and  bankers;  but  as  it  is  inconvenient 
unless  interest  tables  are  used,  it  is  not  generally  adopted. 

Rkmarks. — 1.  Exact  interest,  for  any  period  of  time  expressed  in  days,  may  be  obtained  by 
subtracting  -L  part  from  the  common  interest  for  that  period  of  time. 

2.  Common  interest  may  be  obtained  from  exact  interest  by  adding  thereto  J^  part  of  itself. 

699.  For  convenience,  the  rate  of  interest  should  always  be  expressed  deci- 
mally; the  rules  governing  the  multiplication  and  division  of  decimals  may  then 
be  applied  to  any  product  or  quotient  arising  from  the  use  of  the  decimal  rate. 

Remakks. — 1.  In  many  of  the  States  a  legal  rate  of  interest  is  established,  to  save  dispute 
and  contention  in  cases  of  contracts  in  which  no  rate  of  interest  is  agreed  upon  by  the  parties; 
stiU  the  laws  sanction  an  interest  rate  higher  than  the  fixed  legal  rate,  if  such  rate  be  agreed 
upon  by  the  parties;  in  a  few  of  the  States,  any  rate,  if  agreed  upon,  is  thus  made  legal. 

2.  When  no  particular  rate  of  interest  is  named  in  a  contract  containing  a  general  interest 
clause,  as  "  with  interest,"  or  "  with  use,"  the  legal  rate  of  the  place  where  the  contract  is  made 
is  understood. 

3.  Debts  of  all  kinds  bear  interest  after  they  become  due,  but  not  hefare,  unless  specified. 


SIX   PER   CENT.   METHOD. 

700.  The  following  method  of  computing  interest  is  based  upon  time  as 

usually  reckoned;  i.  e.,  12  months  of   30  days  each,  or  300  days  for  a   year, 

and  is  called  the  Six  Per  Cent.  Metliod.     It  is  convenient  for  use  in  all  cases 

where  time  is  not  given  in  days,  as  for  years  and  months,  or  for  years,  months, 

and  days,  and  where  exact  interest  is  not  required.     Should  the  rate  be  any  other 

than  six  per  cent.,  the  change  can  be  easily  made.     It  is  a  common  method  of 

computing  interest. 

Six  Per  Cent.  Method. 

11.00  in  1  yr.,  at  G,'^,  will  produce  $.06  interest. 
11.00  in  \-  yr.,  or  3  mo.,  at  6^,  will  produce  $.01  interest. 
$1.00  in  1  mo.,  or  30  da.,  at  6^,  will  produce  $  .005  interest. 
$1.00  in  G  da.,  or  ^  mo.,  at  Gf*^,  will  produce  $.001  interest. 
$1.00  in  1  da.,  at  Q^,  will  produce  $.000|  interest. 

701.  To  find  the  Interest  on  Any  Sum  of  Money,  at  Other  Rates  than  6  per 
cent. : 

1.  To  find  the  interest  at  7j^.     Rule. — To  the  interest  at  6'^^  add  otie-sixth 
?/  itself, 

2.  To  find  the  interest  at  7^^.     Rule. — To  the  interest  at  6'^  add  one-fourth- 
of  itself. 


EXAMPLES    IN    INTEREST.  .?11 

5.  To  find  the  interest  at  8^.     Rule. — To  the  interi'st  at  O'i  add  one-third 
of  itself. 

4.     To  find  the  interest  at  9^.     Rule. — To  the  interest  at  6%  add  one-half  of 
itself. 

6.  To  find  the  interest  at  10^.     Rule. — Divide  the  interest  at  6''fo  b>i  >i.  and 
remove  the  decimal  point  one  place  to  the  right. 

6.  To  find  the  interest  at  13^.   Rule, — Multiply  the  interest  at  6^  by  ^. 

7.  To  find  the  interest  at  54^<.  Rule. — From  the  intere.^t  at  64,  subtract 
one-twelfth  of  itself. 

8.  To  find  the  interest  at   b'i.  R»le. — From  the  interest  at  6i,  subtract 
one-sixth  of  itself. 

9.  To  find  the  interest  at  44^^.  Rule. — From  the  interest  at  Si,  subtract 
one-fourth  of  itself. 

10.  To  find  the  interest  at  4^.     Rule. — From  the  interest  at  H':.  sul)tract 
one-third  of  itself. 

11.  To  find  the  interest  at  3^.     Rule. — Divide  the  interest  at  H',  bij  '. 

702.  To  find  the  Interest,  the  Principal,  Rate,  and  Time  being  given. 
Example, — What  is  the  interest  on  $550,  at  6^',  for  3  yr.  8  mo.  12  da.? 

Operation.  Explaxatton.— Since  the  interest  on  .$1  for  1  year  is 

Int.  on  |!l  for  3  yr,    =  $  .  18  $  .OG,  for  3  years  it  will  be  $ .  18;  since  the  interest  on  $1 

"       "        ''8  mo.  =     .04  for  2  months  is  $  .01,  for  8  months  it  will  be  $  .04;  since 

'•-       "        "   13  da.    —     .002  ^^^  interest  on  |1  for  6  da.  is  ,$.001,  for  12  days  it  will 

,               .      „  be  $.002;  therefore  the  interest  on  $1,  at  6  per  cent., 

mt.  on  ^1  lor  6  yr.               ^  ^^^  ^^^  j^jl  ^j^^^^  j^  ^222;  and  the  interest  on  $550  will 

8  mo,  12  da,   ==  $  .222  ^e  550  times  the  interest  on  $1,  or  the  product  of  the  prin- 

$550  X  .222  =  $122.10.  cipal  and  the  rate  for  the  given  time,  which  is  $122.10. 

Rule. — Multiply  the  principal  hij  tlie  decimal  e.vpressiiig  the  interest 
of  one  dollar  for  the  full  time. 

EXAMPLKS  FOR  PRACTICE, 

703.  1.     Find  the  interest  on  $900,  for  4  yr.  1.  mo.  r,  da.,  at  l^L 
Explanation.— Find  the  interest  at  6;^',  and  add  to  it  one-si.xth  of  itself. 

2.  What  is  the  interest  on  $400,  for  1  yr.  7  mo.  2  (hi.,  at  Tij^  ? 
Explanation.— Find  the  interest  at  6^',  and  add  to  it  one-fourth  of  itself. 

3.  What  is  the  interest  on  $150,  for  fi  yr.  3  mo.  IS  da.,  at  8^  ? 
Explanation. — To  the  interest  at  Q%  add  one-third  of  itself. 

If.     Compute  the  interest  on  $1200,  for  3  yr.  4  mo.  15  da.,  at  9^?^. 

Explanation. — To  the  interest  at  6^  add  one-half  of  itself. 

5.     Find  the  interest,  at  10<^,  on  $840,  for  5  yr.  :>  mo.  '.»  da. 

Explanation. — Divide  the  interest  at  6^  by  6,  to  obtain  tlu*  interest  at  \[i,  and  remove  the 
decimal  point  1  place  to  the  right. 


212  EXAMPLES    IN    INTEREST. 

6.  What  is  the  interest,  at  12^,  on  $366,  for  2  yr.  11  mo.  27  da.  ? 
Explanation. — Multiply  the  interest  at  %%  by  2%. 

7.  Find  tlie  interest  on  «!l800,  for  6  yr.  9  mo.  25  da.,  at  5|^. 
Explanation. — Fi-om  the  interest  at  6i  subtract  one-twelfth  of  itself. 

S.     Compute  the  interest,  at  5^,  on  $1000,  for  11  yr.  4  mo.  24  da. 
Explanation.— From  the  interest  at  6^  subtract  one-sixth  of  itself. 

!>.     What  is  the  interest,  at  U^t,  on  $1100,  for  6  yr.  6  mo.  6  da.  ? 
Explanation. — From  the  interest  at  6^'  subtract  one-fourth  of  itself. 

10.  What  is  the  interest,  at  4^,  on  $1350,  for  9  yr.  8  mo.  12  da.  ? 
Explanation. — From  the  interest  at  6^  subtract  one-third  of  itself. 

11.  Find  the  interest,  on  8546,  for  0  yr.  2  mo.  24  da.,  at  3^. 
Explanation. — Divide  the  interest  at  6'^  by  2. 

Remarks. — 1.  Interest  at  any  other  rate,  entire  or  fractional,  can  be  found  by  a  general 
application  of  the  methods  above  explained. 

2.  When  the  mills  of  a  result  are  5  or  more,  add  1  cent;  if  less  than  5,  reject  them. 

12.  Compute  the  interest  on  $752.50,  for  4  yr.  11  mo.  9  da.,  at  6^. 
lo.     Compute  the  interest  on  $3560,  for  9  yr.  10  mo.,  at  8^. 

14.  Compute  the  interest  on  $1540,  for  9  mo.  20  da.,  at  6^. 

lo.  Compute  the  interest  on  $610.15,  for  7  yr.  11  da.,  at  7^. 

IG.  Compute  the  interest  on  $1116,  for  3  yr.  11  mo.  11  da.,  at  5^. 

17.  Compute  the  interest  on  $17500,  for  2  yr.  1  mo.  10  da.,  at  4^^. 

18.  Compute  the  interest  on  $350.40,  for  5  yr.  5  mo.,  at  7^. 

10.     Compute  the  interest  on  $2400,  for  7  yr.  1  mo.  19  da.,  at  10^^. 
20.     Find  the  interest  on  $1450,  from  Aug.  12,  1882,  to  Nov.  10,  1890,  at  6^. 
2 J.     What  is  the  amount  of  $610,  at  8^,  for  3  yr.  8  mo.  21  da.  ? 
Explanation. — The  Principal  plus  the  Interest  equals  the  Amount. 

22.     Find  the  amount  due  after  1  yr.  10  mo.  20  da.,  on  a  6^  loan  of  $1941.50. 

25.  On  the  16th  of  September,  18b4,  I  borrowed  $3500,  at  8,'^  interest.  How 
much  will  settle  the  loan  Jan.  1,  1890? 

24.  After  paying  $225  cash  for  a  horse,  the  purchaser  at  once  sold  him  for 
$275,  on  4  months  credit.     Money  being  worth  7;^,  how  much  was  gained? 

2o.  A  manufacturer  marks  a  carriage  with  two  prices;  the  one  for  a  credit  of 
6  months  on  sales,  and  the  other  for  cash.  If  the  cash  price  was  $750,  and  money 
was  worth  8^,  what  should  ])c  the  credit  price? 

26.  Borrowed  $2750  July  16, 1887,  at  bfo  interest,  and  on  the  same  day  loaned 
it  at  7if«  interest.  If  full  settlement  is  made  Jan.  4,  1889,  how  much  will  be 
gained? 

27.  On  goods  bouglit  for  $4500,  on  6  months  credit,  I  was  offered  5^  off  for 
cash.     If  money  was  worth  6^r,  how  much  did  I  lose  by  accepting  the  credit? 

28.  A  man  sold  his  farm  for  $16000;  the  terms  were,  $4000  cash  on  delivery, 
$5000  in  9  montlis,  $3000  in  1  year  and  six  months,  and  the  remainder  in  2  years 
from  date  of  purchase,  with  6^  interest  on  all  deferred  payments.  What  was 
the  total  amount  paid? 


EXAMPLES    iX    INTEREST. 


213 


29.  May  16th  I  bought  300  barrels  of  flour,  at  *7  i)er  barrel;  July  28th 
I  sold  50  barrels,  at  $8  per  barrel;  Oct.  30th,  100  l>arrels,  at  $6.75  i)er  barrel; 
and  Feb.  13th  following,  the  remainder,  at  S7.80  per  barrel.  Allowing  interest 
at  6^,  what  was  my  gain? 

30.  John  Doe  bought  bills  of  dry  goods  as  follows:  May  3,  ^250;  July  1, 
81125;  Sept.  14,  $450;  Oct.  31,  $150;  Dec.  1st.  $680;  and  on  Dec.  21st,  he  paid 
in  full,  with  6fo  interest.     What  was  the  amount  of  his  payment  ? 

31.  On  March  25,  I  sold  live  bills  of  goods,  for  amounts  as  follows:  S1046.81, 
1952.40,  $173.50,  $1250,  and  $718.25;  and  on  the  first  day  of  the  following 
December  I  received  payment  in  full,  with  interest  at  6'r.  What  was  the 
amount  received? 

32.  A  firm  bought  goods  on  credit,  and  agreed  to  pay  7^  interest  on  each 
purchase  from  its  date;  Oct.  6,  1887,  goods  were  bought  to  the  amount  of  $268  ; 
Dec.  31,  1887,  to  the  amount  of  $765.80;  Feb.  29,  1888,  to  the"amount  of  $600; 
Apr.  1,  1888,  to  the  amount  of  $325.25.  If  full  settlement  was  made  Aug.  25, 
1888,  liow  much  cash  was  paid. 

Remark. — In  the  following  examples,  f^xen  for  teacher's  use  in  class  drill,  the  interest  on 
each  separate  principal  should  be  computed  to  its  nearest  cent;  the  sum  of  the  results  so 
obtained  will  be  the  answer  sought. 

33.  Find  the  amount  of  interest  at  6^,  by  the  six  per  cent,  method, 


On  $680,  for  2  yr.  6  mo.  10  da. 
On  $1895,  for  1  yr  7  mo.  7  da. 
On  $468,  for  5  yr.  5  mo.  1  da. 
On  $1000,  for  11  yr.  1  mo.  20  da. 
On  $645,  for  4  yr.  4  mo.  5  da. 


On  $500,  for  3  yr.  1  mo.  27  da. 
On  $895,  for  5  yr.  11  mo.  11. da. 
On  $1650,  for  1  yr.  10  mo.  23  da. 
On  $1463,  for  9  yr.  1  mo.  9  da. 
On  $365,  for  4  yr.  1  mo.  25  da. 


3Ji..     Find  the  amount  of  interest,  l)y  the  six  jier  cent,  method, 


On  $538,  for  6  yr.  6  mo.  6  da.,  at  9;^. 
On  $1200,  for  7  yr.  4  mo.  27  da.,  at  10^. 


On  $350,  for  3  yr.  7  mo.  18  da.,  at  CH. 
On  8586.50,  for  2  yr.  9  mo.  15  da.,  at  7'i. 
On  $1345,  for  5  yr.  4  mo.  1  da.,  at  8?b. 

35.     Find  the  amount  of  interest,  by  the  six  per  cent,  method, 


t 

On  $675,  for  5  yr.  5  mo.  25  dS,  at  10^. 
On  $1000,  for  llyr.  11  mo.  11  da., at  5;^. 
On  $2500,  for  1  yr.  1  nio.  1  da.,  atlt^^. 
On  $300,  for  2  yr.  2  mo.  2  da.,  at  4^. 
On  $990,  for  4  yr.  4  mo.  6  da.,  at  3f^. 

36.     Find  the  amount  of  interest,  by  the  six  per  cent,  method. 


On  $550,  for  4  yr.  6  mo.  21  da.,  at  Gfc. 
On  $2100,  for  1  yr.  11  mo.  3  da.,  at  7^. 
On  $750,  for  8  yr.  8  mo.  8  da.,  at  S^. 
On  $1200,  for  3  yr.  3  mo.  1  da.,  at  7^^. 
On  $1500,  for  7  yr.  7  mo.  9  da.,  at  9^^. 


On  $250,  for  3  yr.  4  mo.  29  da.,  at  8,^. 
On  $967.25,  fo/7  yr.  0  mo.  27  da.,  at  Qfc. 
On  $1305.09,  forlyr.  11  mo.  7 da., at  7^. 
On  $1255.84,  for  9  mo.  1  da.,  at  lOj^. 
On  $316. 75,  for  5  yr.  1 1  mo.  0  da. ,  at  U^. 
On  $2100. 50,  for  1  yr.  1  mo.  1 9  da. ,  at  9^. 


On  $3546.81,  for  5  yr.  0  mo.  5  da.,  at  3^. 
On  $1867,  for  2  yr.  0  mo.  2  da.,  at  7^^. 
On  $260.60,  for  7  yr.  7  mo.  5  da.,  at  5^. 
On  $1120.95,  for  4  yr.  4  mo.  0  da.,  at  4^. 
On  $1000,  for  5  yr.  6  mo.  7  da.,  at  S^. 
On  $1743,  for  2  yr.  3  mo.  6  da.,  at  6^^. 


• 


214  EXAMPLES    IN    INTEREST. 

704.  To  find  the  Principal,  the  Interest,  Eate,  and  Time  being  given. 

Example.  —  What  principal,  in  3  years  and  2  nionths,  at  6^,  will  gain  $47.50 
interest  ? 

Operation.  Explanation. — Since  $1  in  3  years,  at 

$.18  =  int.  of  $1,  at  G'*',  for  3  yr.  ^  P^^"  ^e°^-'  ^^"  g^'°  ^-^^  '^^^''^''t'  ^^  in  2 
^,         .^      ^A,       ^  n^   £      o'  months  |.01  interest,  it  will  in  the  civen 
.01  =  int.  of  %\;  at  e**,  for  2  mo.  ,.  *  ^^  ■  *    '  ^  •*  *i     •„  • 
'          '  '         time  gain  $.19  interest;  and  if  $1  will  in 

$.19  =  int.  of  $l,at  6jfc,  for  3  yr,  2  mo.       the  given  time  gain  $.19  interest,  the  prin- 

d.1-  -/^  •    i         i.         -^c^        &.n~f\        •  ^  cipal  that  will  in  the  given  time  gain  $47.5*0 

$4  (.oO  interest  ^  .19  =  $2oO,  pr!ncii)al.         .\      ^        ^,  ^         .        ^.-i       * -.a 

'  ^  ^  interest  must  be  as  many  times  $1  as  $.19 

is  contained  times  in  $47.50,  or  $250  ;   therefore  $250  is  the  principal  which  will,  in  3  yr. 

2  mo.,  at  6'V,  gain  $47.50  interest. 

Rule. — Divide  the  given    interest  by  the  iivtrrest  of  one  dollar  for  the 

given  time  and  rate. 

Remark.— "Whenever  the  divLsor  contains  a  fraction  not  reducible  to  a  decimal,  as  in  case 
of  some  fractional  or  odd  ratio  per  cent.,  it  is  better  that  the  fractional  form  be  retained. 
Before  division  in  such  cases,  multiply  both  divisor  and  dividend  by  the  denominator  of  the 
fractional  divisor;  the  relative  value  of  the  terms  will  not  be  changed,  and  greater  exactness 
will  be  secured  in  the  result. 

KXAMPLES   FOR   PKACTICK 

705.  1.    /What  principal,  at  'v',  "will  gain  $154  in  6  yr.  4  mo,  24  da.? 

2.-  What  sum  of  money,  loaned  at  4^^,  for  7  yr.  11  mo.  15  da.,  will  gain 
$1468.21  interest  ? 

3.  "What  sum  of  money,  imested  at  5^^,  will  in  7  yr.  1  mo.  1  da.  produce 
$131.50  interest  ? 

Jf..  A  money  lender  received  $221.68  interest  on  a  sum  loaned  at  8,*^  .July  17, 
1885,  and  paid  Oct.  11,  1888.     What  was  the  sum  loaned  ? 

5.  A  dealer  who  clears  12^^^  annually  on  his  investment,  is  forced  by  ill  health 
to  give  up  his  business;  he  lends  his  money  at  7^,  by  which  his  income  is  reduced 
$1512.50.     How  much  had  he  invested  in  his  business  ? 

6.  How  many  dollars  mitst  I  put  at  interest,  at  9^,  Jan.  27,  1889,  .so  that  on 
the  18th#>f  Xov..  1895,  $506.27  interest  will  be  due? 

706.  To  find  the  Principal,  the  Amount,  Rate,  and  Time  being  given. 

Ex.v.MPLK. — What   i>rincipal,  at  ij'ft,,  will,   in   4  yr.  G  mo.    15   da.,  amount  to 

$2372.25? 

Operation.  Explanation.  —  Since  a  principal  of 

4,,  or/.T-  t.     e  S.A  r^t\  e       l\      j.-  $1  "^^^h   in   the    given  time,   amount    to 

$1.272o  =  amount  of  $1.00  for  the  time.     ^.  o-o-   •.      -n  •  •     •    i    <• 

$1.272.D,  It  will  require  a  principal  of  as 

$2372.25  -f-  1,2725  =  $1800,  principal.  many  times  $1  to  amount  to  $2372.25  as 

$1.2725  is  contained  times  in  $2372.25,  or 

$1800. 

Rule. — Divide  the  uimmnt  by  tlir  amount  of  1  dollar  for  the  given 
time  and  rate. 


EXAMPLES    IX   INTEREST.  215 

EXAMPLES   FOK  PRACTICE. 

707.  1-  What  Slim,  put  at  interest  at  '7'fc  for  5  yr.  11  mo.  3  da.,  will  amount 
to  $630.90? 

2.  A  boy  is  now  15  years  old.  How  much  must  be  invested  for  him,  at  7^5^ 
simple  interest,  that  he  may  have  $15000  when  he  becomes  of  age  ? 

3.  What  sum,  put  at  interest  June  1,  1888,  at  7^,  will  amount  to  $687.50 

July  1,  1890? 

Jf.      What  sum  of  money,  put  at  interest  to-day  at  5^,  will  amount  to  $1031.25 

in  7  mo.  15  da.  ? 

5.  What  principal  will  amount  to  $308.34:  in  11  mo.  9  da.,  at  6j^  ? 

6.  A  man  loaned  a  sum  of  money  to  a  friend  from  June  13  to  Dec,  1,  at 
7^  when  he  received  $763.28  in  full  payment.     How  much  was  loaned  ? 

7.  Owing  a  debt  of  $2146.18,  due  in  1  yr.  7  mo.  18  da.,  I  deposited  in  a  bank, 
allowing  me  6^  interest,  a  sum  sufficient  to  cancel  my  debt  when  due.  Find  the 
sum  deposited. 

708.  To  find  the  Eate  Per  Cent.,  the  Principal,  Interest,  and  Time  being  given. 

Example. — At  what  rate  per  cent,  must  $750  be  loaned,  for  2  yr.  5  mo.  6  da., 
to  gain  $164.25  interest  ? 

Operation.  Explanation. — The  principal  will  gain 

.  ^  „        ,      , .  ,  ^  .         $18.25  interest  in  the  given  time  at  1  per 

$18.25  =  ]nt.  of  $750  for  the  time  at  1^.       cent. ;  in  order  that  it  may  in  the  given  time 

$164.25  -7-  $18.25  _  9  or  9^.  ^..^^^  $164.25,  the  rate  must  be  as  many 

times  1  per  cent,  as  $18.25  is  contained 
times  in  $164.25,  or  9  per  cent. 

Rule. — Divide  the  given,  irvterest  hy  the  interest  on  tlie  given  principal 
for  the  given  time,  at  1  per  cent. 

Remark. — When  the  amoimt,  interest,  and  time  are  given,  to  find  the  rate  percent.,  subtract 
the  interest  from  the  amount,  thus  finding  the  principal,  then  proceed  as  by  the  above  rule. 

EXAMPLES   FOK   PRACTICE. 

709.  1.  If  I  pay  $518.75  interest  on  $1250,  for  5  yr.  0  mo.  12  da.,  what  is 
the  rate  per  cent.  ? 

2.  At  what  rate  would  $710,  in  3  yr.  5  mo.  20  da.,  produce  $172.56  interest  ? 

3.  At  what  rate  would  $4187.50  amount  to  $4738.68,  in  1  yr.  11  mo.  12  da.? 
i.     If  $1200  amounts  to  $2135.80  in  12  yr.  11  mo.  29  da.,  what  is  the  rate 

per  cent.  ? 

6.  A  lady  deposited  in  a  savings  bank  $3750,  on  which  she  received  $93.75 
interest  semi-annually.    What  per  cent,  of  interest  did  she  receive  on  her  money? 

6.  A  debt  of  $480,  with  interest  from  August  24,  1886,  to  Dec.  18,  1888, 
amounted  to  $546.72.     What  was  the  rate  per  cent,  of  interest  ? 

7.  To  satisfy  a  debt  of  $1216.80,  that  had  been  on  interest  for  4  yr.  4  mo. 
21  da.,  I  gave  my  check  for  $1751.18.     What  was  the  rate  per  cent,  of  interest? 


216  SHORT  METHODS   FOR   FINDING   INTEREST. 

710.  To  find  the  Time,  the  Principal,  Interest,  and  Rate  being  given. 

Example. — In  what  time  will  $540  gain  |i74.52  interest,  at  6^  ? 

Opekation.  Explaxatiox. — Since  in  1  year  $540  will,  at. 

$32,40  =  int.  on  $540  for  1  vr.,  at  6<.      ^  P^'"  cent., gain  #32.40  interest,  it  will  require 

f^A  5->  _i_  •^•)  40  —  o  Q              ~  '         as  many  years  for  it  to  gain  .$74.52  interest  as 

*    ^    '      ^*       .-)  Q~"    '  f32.40  is  contained  times  in  $74.52,  or  2.3  years; 

Z:6  X  1  yr.  =  :..3  years.  find,  by  the  rule  for  the  reduction  of  a  denomi- 

.3  yr.   X  12  =  3.6  months.  nate  decimal,  that  2.3  years  equals  2  yr.  3  mo. 

.6  mo.  X  30  =  18  days.  18  da. 

2  yr.  3  mo.  18  da. 

Remark. — When  by  inspection  it  is  apparent  that  the  time  is  less  than  a  year,  divide  the 
given  interest  by  the  interest  on  the  principal  for  the  highest  apparent  unit  of  time;  the  quotient 
will  be  in  units  of  the  order  taken,  which  reduce  as  above. 

Rule. — Divide  the  given  interest  hy  the  interest  on  the  princijxil  for  1 

year,  at  the  given  rate  ])rr  eent. 

Remark. — When  the  amount,  interest,  and  rate  are  given  to  find  the  time,  subtract  the 
interest  from  the  amount,  thus  finding  the  principal,  and  proceed  as  above. 

EXABIPLES   FOR   PRACTICE. 

711.  1.     How  long  will  it  take  $360  to  gain  $53.64,  it  6^. 

2.  How  long  should  I  keep  $466.25,  at  8fr,  to  have  it  amotint  to  $610.48  ? 

3.  A  debt  of  $1650  was  paid,  with  bl'/c  interest,  on  Aug.  30,  1888,  by  deliver- 
ing a  check  for  $2316.85.     At  what  date  was  the  debt  contracted  ? 

4.  How  long  must  $612  be  on  interest,  at  7,^^',  to  amount  to  $651.27  ? 

5.  On  April  1,  1888,  I  loaned  $1120,  at  5^,  and  when  the  money  was  due  I 
received  $1202.60  in  full  payment.     What  was  the  date  of  the  payment  ? 

6.  In  what  time  will  money,  bearing  8^  simple  interest,  double  itself  ? 

ExPLAXATiox. — In  order  to  double  itself,  the  interest  accumulated  must  be  equal  to  the 
principal,  or  be  100  per  cent,  of  the  principal.  And  since  the  principal  increases  8  per  cent,  in 
one  year,  it  will  require  as  many  years  to  increase  100  per  cent.,  or  to  double  itself,  as  8  per 
cent,  is  contained  times  in  100  per  cent.,  or  124,  equal  to  12  yr.  6  mo. 


SHORT   METHODS   FOR   FINDING    INTEREST. 

712.     To  find  Interest  for  Days,  at  6  per  cent.,  360  day  basis,  or  Common  Interest. 

ExPLAXATiox. — A  principal  of  |1  will,  in  1  year,  at  6  per  cent.,  gain  $.06  interest.  A  prin- 
cipal of  $1  will,  in  I  year,  or  2  months,  or  60  days,  at  6  per  cent.,  gain  .01  interest.  Since 
$.01  equals  j-J^  of  the  principal,  the  interest  on  any  sura  of  money  for  60  days,  at  6  per  cent., 
can  be  found  by  pointing  off  two  integral  places  from  the  right;  and  since  6  is  y'g  of  60,  the 
interest  for  6  days  Ciin  be  found  by  pointing  off  three  places;  and  since  ten  times  60  is  600,  the 
interest  for  600  days  is  ten  times  that  for  60  days,  and  may  be  found  by  pointing  off  1  place; 
and  since  6000  is  ten  times  600,  the  interest  for  6000  days  can  be  found  by  multiplying  the  inter- 
est for  600  days  by  10,  or  in  other  words,  the  interest  for  6000  days  will  equal  the  principal; 
the  principal  thus  being  shown  to  double  itself  in  that  time  at  6  per  cent.  This  may  further 
be  proved  true  from  either  of  two  illustrations: 


EXAMPLES    IN    FlXDINft   IXTEREST.  :^17 

1st.     6000  da,  -^  :JGO  (12  X  30)  =  16|,  or  16  yr.  +  8  nio. 

2d.     lOOfc  -^  6;c  =  16^,  or  16  yr.  +  8  mo. 

Hence,  assuming  $3136  as  a  principal,  we  form  the  following 

Table. 

%2136  =  principal. 

12.136  =  interest  at  Qfo  for  6  days. 

$21.36  =  interest  at  6^  for  60  days. 

$213.6  =  interest  at  ^  for  600  days. 

$2136.=  interest  at  6^  for  6000  days. 

Remakks. — 1.  Observe,  as  above  stated,  that  the  interest  for  6000  days  equals  the  principal, 
or  that  anj-  sura  of  money  will,  at  common  interest,  double  itself  in  6000  days. 

2.  Since  interest  is  ordinarily  computed  on  the  basis  of  860  days,  or  12  periods  of  30  days 
each,  as  illustrated  above,  all  results  will  be  required  on  that  basis,  unless  otherwise  specified. 

713. — 1.  To  find  the  interest  of  any  sum  of  money,  at  ^4,  for  6  days. 
Bulk.  —  Cut  off  three  integral  ;placesfrom  the  right  of  the  principal. 

2.  To  find  the  interest  of  any  sum  of  money,  at  6^  for  60  days.  Rule. — Cut 
off  two  integral  places  from  the  right  of  the  j)rincipal. 

3.  To  find  the  interest  of  any  sum  of  money,  at  (j'fc,  for  600  days.  Rule. — Cut 
off'  one  integral  place  from  the  right  of  the  principal. 

Jf.  To  find  the  interest  of  any  sum  of  money,  at  6,^,  for  6000  days.  Rule. — 
Write  the  interest  as  being  equal  to  the  2^rincipal. 

Remark. — Interest  is  a  product  of  which  the  rate  and  time  are  factors.  [Formula. — Interest 
=r  Principal  X  Rate  X  Time.]  Since  the  rate,  being  a  constant  factor,  may  be  ignored,  it  will 
be  observed  that  it  will  make  no  difference  if,  for  convenience,  the  term  principal  (in  dollars),  and 
that  of  time  (in  days),  be  interchanged.  Illustration:  The  interest  of  500  (dollars)  for  93  (days), 
is  the  same  as  the  interest  of  93  (dollars)  for  500  (days);  and  since  500  is  ^^j  of  6000,  the  interest 
required  can  be  found  by  dividing  93  (dollars)  by  12,  which  gives  $7.75.  Again,  the  interest 
of  150  (dollars)  for  S8  (days)  equals  the  interest  of  88  (dollars)  for  150  (days);  and  since  150  is 
i  of  600,  the  required  interest  is  obtained  by  pointing  off  one  place  from  the  right  of  88  (dollars), 
as,  $8.8,  and  dividing  the  result  by  4,  obtaining  $2.2,  or  $2.20,  as  the  interest. 

714.     To  find  Interest  at  Other  Rates  than  6  per  cent.,  360  Day  Basis. 

1.  To  find  the  interest  on  any  sum  of  money  for  12  days,  at  6  per  cent. 
Rule. — Point  off'  3  ^jlaces  and  multi])li/  by  2. 

Remakks. — 1.     For  any  number  of  days  divisible  by  6,  proceed  in  like  manner. 

2.  For  other  rates,  add  or  subtract  fractional  parts  of  results,  as  in  Art.  701. 

3.  For  odd  days,  add  fractional  parts  to  the  result. 

2.  To   find  the  interest  for  18  days,  at  7;k     Rule. — Point  off'  3  places , 
inultiply  by  3,  and  to  the  result  add  one-sixth  of  itself. 

3.  To  find  the   interest  for  24   days,  at   o^L     Rule. — Point  off  3  places^ 
mziltiply  by  4,  and  from  the  result  subtract  one-sixth  of  itself. 

4.  To  find  the  interest  for  36  days,  at  4^^.     Rule. — Point  off  3  places, 
rmiltijily  by  6,  and  from  the  result  subtract  one-fourth  of  itself. 

5.  To  find   tiie  interest  for  78  days,   at  %<fc.     Rile. — Point  off'  3  places, 
multiply  by  13,  and  to  the  result  add  one-third  (f  itself. 


218 


EXAMPLES   IN   INTEREST. 


6.  To  find  the   interest  for  51  days,  at  6^.     Eule. — Point  off  S  places, 
multiply  hy  8,  and  to  the  result  add  one-half  of  the  first  result. 

Remark. — In  a  similar  way  all  changes  of  time  and  rate  may  be  considered. 

7.  To  find  the  interest  for  10  days,  at  6^.     Rule. — Point  off  2  places,  and 
divide  the  result  by  6. 

8.  To  find  the  interest  for  20  days,  at  1^.     Rule. — Poitit  off  2  places,  divide 
the  result  hy  3,  and  to  the  quotient  add  one-sixth  of  itself. 

9.  To  find  the  interest  for  30  days,  at  7^^.     Rule. — Point  off  2  jda^es, 
divide  the  result  hy  2,  and  to  the  quotient  add  one-fourth  of  itself  . 

10^  To  find  the  interest  for  40  days,  at  9j^.  Rule. — Point  off  2  places,  sub- 
tracirfrom  the  result  one-third  of  itself,  and  to  the  remainder  add  one-half  of  itself. 

11.  To  find  the  interest  for  45  days,  at  8^.  Rule. — Point  off  2  places,  sub- 
tract from  the  result  one-fourth  of  itself,  and  to  the  remainder  add  one-third  of 
itself 

12.  To  find  the  interest  for  54  days,  at  6 
from  the  residt  subtract  otie-tenth  of  itself 

IS.     To  find  the  interest  for  240  days,  at 
■multiply  by  4- 


Rule. — Point  off  2  places,  and 
Rule. — Point  off  2  places  and 


Remarks. — In  a  similar  manner  obtain  interest  for  all  terms  of  60  days  or  parts  thereof ,  and 
at  any  required  rate. 


IJf-.     To  find  the  interest  for  50  days,  at  6^. 
divide  by  12. 

15.  To  find  the  interest  for  100  days,  at  Q,^L 
divide  by  6. 

16,  To  find  the  interest  for  150  days,  at  Qfji. 
divide  by  4- 


Rule. — Point  off  1  place  and 
Rule. — Point  off'  1  place  and 
Rule. — Point  off  1  place  and 


Remark. — Daily  cla.ss  drill  for  five  or  ten  minutes,  during  the  time  given  to  the  subject  of 
interest  and  its  varied  applications,  will  impart  to  the  class  an  astonishing  degree  of  accuracy 
and  rapidity  in  computing  interest;  and  while  odd  rates  are  not  in  common  ase,  valuable  drill 
may  be  given  by  their  occasional  introduction,  and  the  varied  changes  necessary  to  obtain 
interest  for  odd  days  will  insure  the  very  best  results. 


EXAMPLES  FOR    PRACTICE. 


715.  Find  the  interest  on 

1.  $1750,  for  15  days,  at  6^. 

2.  $1125,  for  24  days,  at  7^. 

3.  $742.50,  for  30  days,  at  6^. 

4.  $900,  for  93  days,  at  l\i. 
^.  $G60,  for  63  days,  at  8^. 

fj.  $136.42,  for  33  days,  at  9^. 

7.  $1000,  for  21  days,  at  10^. 

8.  $2000,  for  12  days,  at  h^. 

9.  $351.23,  for  40  days,  at  4^. 
10.  $1368,  for  50  days,  at  3^. 


11.  $93.40,  for  150  days,  at  6^. 

12.  $550,  for  75  days,  at  7^. 

13.  $842.50,  for  45  days,  at  6^. 
IJf.  $800,  for  27  days,  at  5^. 

15.  $1725,  for  57  days,  at  9^. 

16.  $125,  for  55  (fays,  at  6^. 

17.  $3741.85,  for  6  days,  at  7^. 

18.  $5178,  for  9  days,  at  9^. 

19.  $732,  for  11  days,  at  6j^. 

20.  $1174.51,  for  42  days,  at  8^ 


EXAMPLES   FOU    PRACTICE. 


219 


$120,  for  49  days,  at  9^. 
160,  for  50  days,  at  5^. 
1930,  for  83  days,  at  6^. 
1750,  for  84  days,  at  6^. 
$550,  for  72  days,  at  7j^. 
166.90,  for  11  days,  at  6^. 
$83.21,  for  30  days,  at  Oy^. 
$110.25,  for  60  days,  at  7^. 
$77.54,  for  54  days,  at  6^. 
$300,  for  66  days,  at  10^. 
$800,  for  93  days,  at  8^. 
$1110,  for  63  days,  at  6^. 
$684,  for  50  days,  at  6fi. 
$1250,  for  70  days,  at  12^'. 
$351.89,  for  9  days,  at  6^. 

Remark. — In  the  five  following  examples,  compute  the  interest  on  each  separate  principal 
to  the  nearest  cent;  then  find  the  sum  total  of  the  interest  thus  obtained. 


31. 

$340,  for  70  days,  at  10^. 

36 

23. 

$1478,  for  80  days,  at  6^. 

37. 

2S. 

$2150,  for  96  days,  at  4^^. 

38. 

21 

$1200,  for  53  days,  at  6^. 

39. 

26. 

$1500,  for  87  days,  at  7^. 

40 

26. 

$420,  for  41  days,  at  5^. 

41. 

27. 

$360,  for  81  days,  at  6^. 

43. 

28. 

$2347.50,  for  18  days,  at  7^. 

¥i 

29. 

$1112.49,  for  25  days,  at  8^. 

u 

30. 

$1300,  for  13  days,  at  6^. 

45. 

31. 

$17000,  for  3  days,  at  5^^. 

46. 

32. 

$195.50,  for  33  days,  at  10^. 

A7. 

33. 

$1050,  for  43  days,  at  7^. 

48. 

34. 

$1560,  for  44  days,  at  7i^. 

49. 

35. 

$180,  for  47  days,  at  &'fr. 

50 

716.     1.     Find  the  total  amount  of  interest  on 


$550,  for  18  days,  at  6,<.  - 
$810,  for  40  days,  at  7^. 
$1000,  for  41  days,  at  74%'. 
$342.50,  for  42  days,  at  5<i. 
$1362.50,  for  45  days,  at  6f/. 


$250,  for  50  days,  at  6^. 
$593.25,  for  80  days,  at  7^. 
$1966,  for  75  days,  at  b^. 
$450,  for  83  days,  at  8^. 
$990,  for  63  days,  at  6^. 


2.  Find  the  total  amount  of  interest  on 
$720,  for  9  days,  at  10^. 
$7500,  for  3  days,  at  7^. 
$216,  for  93  days,  at  8^. 
$504,  for  54  days,  at  6^. 
$600,  for  4  days,  at  4^%'. 


$1124,  for  15  days,  at  3j 
$550,  for  45  days,  at  7^^ 
$160,  for  27  days,  at  6^. 
$240,  for  31  days,  at  8^. 
$540,  for  41  days,  at  9^. 


S.     Find  the  total  amount  of  tlie  interest  on 


)2,  for  8  days,  at  3^. 
$1728,  for  10  days,  at  6^. 
$2150.42,  for  17  days,  at  7^. 
$519,  for  24  days,  at  8%. 
$1600,  for  23  days,  at  74^. 


Find  the  total  amount  of  interest  on 
$695,  for  *79  days,  at  3^. 
$546,  for  73  days,  at  ZH. 
$1382.50,  for  69  days,  at  4^. 
$101.80,  for  65  days,  at  Ui.      ' 
$500,  for  61  days,  at  5<. 


$1400,  for  26  days,  at  6^ 
$1700,  for  29  days,  at  8^ 
$1900,  for  37  days,  at  7^ 
$2100,  for  43  days,  at  6^ 
$3100,  for  53  days,  at  3^ 


$99,  for  59  days,  at  5^^. 
$780,  for  101  days,  at  6fK 
$1350,  for  150  days,  at  Q^'i. 
$775,  for  180  days,  at  7^.' 
$938.20.  for  10  days,  at  10,^. 


220 


EXAMPLES   FOR    PRACTICE. 


5.     Find  the  total  amount  of  interest  on 

$285.56,  for  11  days,  at  11^. 
$372.40,  for  21  days,  at  7^. 
$519.31,  for  27  days,  at  7^. 
$3000,  for  1  day,  at  6,^. 
$6000,  for  5  days,  at  5^. 


$10000,  for  16  days,  at  8^. 
$400,  for  48  days,  at  6^. 
$2400,  for  54  days,  at  5^, 
$730.30,  for  33  days,  at  9^. 
$100,  for  45  days,  "at  6^. 


717.     To  find  Interest  for  Days  at  6  per  cent,  365  day  basis,  or  Exact  Interest. 

Remarks. — 1.  Aside  from  uses  in  government  calculations,  exact  interest  is  rarely  com- 
puted; and  while  it  is  enforceable,  being  strictly  legal,  the  greater  convenience  of  the  360  day 
rules  so  commend  them  to  public  favor  as  to  lead  to  their  common  use. 

2.  On  a  basis  of  12  periods  of  30  days  each,  or  360  days  for  a  year,  the  year's  interest  is 
taken  for  a  period  too  short,  since  the  year  (exclusive  of  leap  year)  contains  365  days.  The 
time  is,  therefore,  5  days  or  ^%^,  equal  to  ^-^,  too  short,  and  the  interest  taken  on  that  basis  is 
proportionally  too  great;  to  correct  this  error  and  obtain  the  exact  interest,  subtract  y'j  part 
from  any  interest  obtained  on  a  360  day  basis. 

EXAMPI.ES  FOR  PRACTICE. 


718.     1.     Find  the  exact  in 

~.  Find  the  exact  interest 

3.  Find  the  exact  interest 

4-  Find  the  exact  interest 

'J.  Find  the  exact  interest 

6.  Find  the  exact  interest 

7.  Find  the  exact  interest 

8.  Find  the  exact  interest 
•9.  Find  the  exact  interest 

10.  Find  the  exact  interest 


terest  of  $630,  for  50  days,  at  6^. 
of  1954,  for  63  days,  at  7^. 
of  $800,  for  33  days,  at  5^. 
of  $137.50,  for  93  days,  at  8^. 
of  $210.54,  for  100  days,  at  9^. 
of  $681.80  for  90  days,  at  10^. 
of  $500,  for  48  days,  at  6^. 
of  $1200,  for  31  days,  at  hi. 
of  $1500,  for  55  days,  at  7i^. 
of  $811.25,  for  45  days,  at  4^^. 


Remark. — In  the  three  following  examples,  find  the  exact  interest  on  each  separate  prin- 
cipal to  the  nearest  cent,  and  then  the  total  of  the  interest  thus  obtained. 


719.     1. 


ADDITIONAL    EXAMPLES    FOR    PRACTICE. 

Find  the  total  amount  of  exact  interest  on 


$510,  for  63  days,  at  7^. 
$615,  for  93  days,  at  6^. 
$450,  for  78  days,  at  5^. 
$120,  for  96  days,  at  7^^. 
$353,  for  80  days,  at  10^. 


$1935.60,  for  75  days,  at  hi. 
$2136.88,  for  70  days,  at  4^. 
$1000,  for  73  days,  at  6^. 
$2000,  for  146  days,  at  9^. 
$1500,  for  219  days,  at  4^. 


2.     Find  the  total  amount  of  exact  interest  on 


$2150,  for  65  days,  at  3^. 
$1640,  for  14  days,  at  4^. 
$900,  for  17  days,  at  m. 
$182.79,  for  24  days,  at  5^. 
$605.51,  for  33  days,  at  6^. 


$890.90,  for  45  days,  at  7|^. 
$1100,  for  46  days,  at  8,*^. 
$2500,  for  54  days,  at  10^. 
$720,  for  66  days,  at  9^. 
^365,  for  51  days,  at  6^. 


PERIODIC    INTEREST.  2'il 

"Find  the  total  amount  of  exact  interest  on 


$96.60,  for  20  days,  at  7^. 
$138.24,  for  15  days,  at  6^. 
$1793.80,  for  35  days,  at  8^. 
$2000,  for  7  days,  at  7^. 
$1000,  for  1  day,  at  4^^ 


$615.62,  for  93  days,  at  6^. 
$730,  for  57  days,  at  5^. 
$891.11,  for  63  days,  at  6^. 
$200,  for  10  days,  at  6^4. 
$525,  for  25  duvs,  at  10%. 


PERIODIC    INTEREST. 

720.  Annual  Interest  is  simple  interest  on  the  principal  for  each  year 
j)eriod,  and  on  each  year's  interest  remaining  unpaid. 

721.  Semi- Annual  Interest  is  simjjle  interest  on  the  principal  for  eacli  half- 
year  period,  and  on  each  period's  interest  remaining  unpaid. 

722.  Quarterly  Interest  is  simjyle  interest  on  the  principal  for  each  quar- 
ter-year period,  and  on  each  period's  interest  remaining  unpaid. 

723.  In  some  States  annual  and  other  periodic  interest  is  sanctioned  by  law; 
but  in  many  States  it  cannot  be  legally  enforced. 

724.  When  the  interest  payments  are  not  made  when  due,  periodic  interest 
becomes  greater  than  siinple  interest,  because  of  the  interest  on  the  unpaid  sums. 

725.  Perodic  interest  is  sometimes  secured  by  a  note  or  series  of  notes;  in 
such  cases  the  principal  only  is  secured  by  one  of  the  series  (if  not  by  mortgage 
or  otherwise),  while  each  of  the  other  notes-  is  drawn  for  one  interest  payment, 
and  matures  on  the  date  at  which  such  payment  is  due.  By  such  arrangement, 
periodic  interest  can  be  enforced  in  States  where  it  would  otherwise  be  regarded 
as  illegal. 

726.  In  States  where  })eriodic  interest  is  legal,  the  contract  should  contain 
the  words,  "with  annual  interest,"  or  "  with  interest  payable  annually,"  or  "  with 
semi-annual  interest,"  etc. 

727.  As  simple  interest  cannot  be  collected  until  the  principal  is  due,  simple 
and  periodic  interest  are  the  same  up  to  the  end  of  the  first  interest  period. 

Remark. — When  the  interest  is  not  paid  at  the  end  of  the  periods,  as  agreed,  much  time 
will  be  saved  in  obtaining  the  amount  due,  by  finding  the  interest  on  one  over-due  payment 
for  the  aggregate  of  the  time  for  which  they  were  all  over-due;  to  this  interest  add  the  amount 
of  the  principal,  at  simple  interest. 

728.  To  find  Periodic  Interest,  the  Principal,  Rate,  and  Time,  being  given. 

Example. — What  is  the  interest  on  $2500,  from  July  1,  1885,  to  Sept.  16, 
1888,  at  6^  interest,  due  annually,  and  no  payments  miide  until  final  settlement? 


222  COMPOUND   INTEREST. 

Operation.  Explanation. — From    July  1, 

1888 — 9—16  1885,  to  Sept.  16,  1888,  is  3  yr.  2 

Iggg <^ 1  mo.    15   da.     And   since   the  first 

; .  j'car's  interest,  which  is   $150,  was 

^~^~1^  =  *^°^^-                 _  not  paid  until  2  yr.  2  mo.  15  da. 

$2500  X  .06  =  $150  =  1  yr.  int.  after  it  was  due,  the  second  year's 

-r,         .    .  •  ■,  (  2  yr.  3  mo.  15  da.  interest,  $150,  was  not  paid  until 

Remaining  unpaid  \  ^  ^^^  ^  ^^^  ^^  ^^  ^  ^^^  ^  ^^  ^^  ^^  ^^^^^.  .^  ^,^  ^^^^ 

101  perioas  oi        ^  2  mo.  15  da.  and  the  third  year's  interest,  $150, 

Interest  of  $150.  for  3  yr.  7  mo.  15  da.  =  $  32.63  ^^'^s  not  paid  until  2  mo.  15  da. 

o  ■       1     •    i.        i.  •    ^-    „i  ^;^A  AA  after  it  was  due,  the  aggregate  of 

3  yr.  Simple  interest  on  principal  =    4o0.00  •      .      ^       ,.,   ■  .  °      ,     ,j 

''  '         .  .      .      ,  ^1    ^-  the  time  for  Avhich  interest  should 

2  mo.  15  da.  interest  on  principal  =      o\.%o  ^^  computed  on  one  year's  interest. 

Total  interest  due  —  $513.88     $150,  is  3  yr.  7  mo.  15  da.,  and  its 

interest  for  that  time  is  $32.63. 
Adding  to  this  the  interest  of  the  principal  for  the  full  time,  $481.25,  gives  $513.88,  the  amount 
of  interest  due. 

Rule.— To  the  svjnple  interest  on  the  principal  for  the  full  time,  add 
the  interest  on  one  period's  interest  for  the  aggregate  of  time  for  irhich 
the  payments  of  interest  were  deferred. 

EXAMPLES   FOR   PRACTICE. 

729.  1.  What  is  the  annual  interest  of  $1260,  payments  due  semi-annually 
from  May  21,  1884,  to  Nov.  9,  1888,  at  7j^,  no  interest  having  been  paid? 

2.  What  is  the  annual  interest  of  $3416.50,  ijuynients  due  quarterly  from 
Jan.  15,  1882,  to  Sept.  6,  1889,  at  b'/c,  no  interest  having  been  paid? 

3.  Find  the  amount  of  interest  due  at  the  end  of  4  yr.  9  mo.  on  a  note  for 
$1155,  at  6^,  interest  payable  annually,  but  remaining  unpaid. 

Jf.  On  a  note  of  $1750,  dated  Aug.  1,  1882,  given  with  interest  payable 
annually  at  10^,  the  first  three  payments  were  made  when  due.  How  much 
remained  unpaid,  debt  and  interest,  Jan.  1,  1889? 

5.  Find  the  amount  due  Oct.  11,  1891,  on  a  debt  of  $11000  under  date  of 
July  5,  1888,  bearing  4^^  interest,  payable  quarterly,  notes  for  the  quarterly 
interest  having  been  given  and  nothing  paid  until  final  settlement. 


COMPOUND    INTEREST. 

730.  Compound  Interest  is  the  interest  on  the  i)rincip;il  and  on  tlie  unimid 
interest  after  it  l)ecomes  due. 

731.  The  Simple  Interest  may  be  added  to  the  principal  annually,  semi- 
annually, quarterly,  or  for  other  agreed  periods;  when  done,  interest  is  said  to 
be  compounded  annually,  quarterly,  etc.,  as  tlie  case  may  be. 

732.  General  Rule. — Find  the  amount  of  the  prinei])<tl  and  i interest 
for  the  first  period,  and  mahe  that  the  principal  for  the  second  period, 
and  so  proceed  to  the  time  of  settlement. 


COMPOUND   INTEREST. 


223 


Remarks 1.  If  the  time  contains  fractional  parts  of  a  period,  as  months  and  days,  find 

the  amount  due  for  the  full  periods,  and  to  this  add  its  interest  for  the  months  and  days. 

2.  Compound  interest  is  not  recoverable  by  law,  but  a  creditor  may  receive  it  if  tendered, 
■without  incurring  the  penalty  of  usury;  a  new  obligation  may  be  taken  at  the  maturity  of  a 
compound  interest  claim,  for  the  amount  so  shown  to  be  due,  and  such  new  obligation  will  be 
valid  and  binding. 

733.  To  find  the  Compound  Interest,  when  the  Principal,  Rate,  and  Time  of 
Computing  it  are  given. 

Example.— Find  the  interest  of  $750,  for  3  yr.  8  mo.  15  du.,  at  6^,  if  interest 
be  compounded  annually. 

Operation. 

$45         =  int.  for  1st  yr. 

$795       =  amt.  at  end  of  1st  yr. 

$47.70    =  int.  for  2d  yr. 

$842. 70  =  amt.  at  end  of  2d  yr. 

$50.56    =int.  for  3d  yr. 

$893.26  =  amt.  at  end  of  3d  yr. 

$38.16    =  int.  for  8  mo.  15  da. 

$931.42  =  amt.  for  full  time. 

$931.42  —  $750  =  $181.42,  comp.  int.  full  time. 


Explanation. — Since  the  interest 
is  to  be  compounded  annually,  the 
amount  due  at  the  end  of  the  first 
year,  which  is  $795,  will  be  the  basis 
of  the  interest  for  the  second  year; 
and  the  amount  due  at  the  end  of  the 
second  year,  $842.70,  will  be  the  basis 
of  the  interest  for  the  third  year;  the 
amount  due  at  the  end  of  the  third 
year,  $893  26,  will  be  the  basis  of  the 
interest  for  the  remaining  8  mo.  15 
da.  of  the  time;  and  since  the  com- 
pound amount  thus  found,  $931.42, 

is  made  up  of  the  compound  interest  and  the  principal,  if  from  this  amount  the  principal  be 

subtracted,  the  remainder,  $181.42,  will  be  the  compound  interest. 

Rule.— I.  Find  the  amount  on  the  principal  for  the  first  interest  period; 
take  this  result  as  a  principal  for  the  next  period,  and  so  on  through  the 
whole  time. 

II.  Subtract  the  principal  from  the  last  amount,  and  the  remainder 
will  he  the  com/pound  interest. 

Remark. — For  half  or  quarter  years,  take  one-half  or  one-quarter  the  rate  per  cent,  for  one 
year. 

EXAMPLES  FOR  PRACTICE. 

734.  1.  What  is  the  compound  interest  on  $1200,  for  4  years,  at  7^,  if  the 
interest  is  compounded  annually? 

2.  What  is  the  compound  interest  on  $600,  for  3  years,  at  h<;i,  if  the  interest 
is  compounded  quarterly? 

3.  Wliat  is  the  compound  interest  on  $1640,  for  2  yr.  6  mo.,  at  lOf^,  if  the 
interest  is  compounded  quarterly? 

4.  Find  the  compound  interest  on  $1000,  for  4  yr.  5  mo.  12  da.,  at  8f^,  if  the 
interest  is  compounded  sei^iii-annually  ? 

5.  What  will  be  the  amount  due  Feb.  11,  1892,  on  a  debt  of  $900,  bearing 
8^  interest,  compounded  quarterly,  if  the  debt  bears  interest  from  July  1,  1888  ? 

6.  Oct.  1,  1888,  I  paid  in  full  a  note  for  $1350,  dated  March  15,  1883,  and 
bearing  10^  interest.  If  the  interest  was  compounded  semi-annually,  what  was 
the  amount  due  at  settlement  ? 


224 


COMPOUND    INTEREST   TABLE. 


735.     The  labor  of  cominiting  compound  interest  may  be  greatly  shortened 
by  the  use  of  the  following 

Conipoiiud    Interest    Table, 

Showing  the  amount  of  ^1  at  compound  interest  at  various  rates  per  cent, 
for  liny  number  of  years,  from  1  year  to  50  years,  inclusive. 


Yrs. 

1  per  ct. 

1}4  per  ct. 

2  ])er  ct. 

214  per  ct. 

3  per  ct. 

3K  per  ct. 

4  per  ct. 

1 

1.0100  000 

1.0150  000 

1.0200  0000 

1.0250  0000 

1.0300  0000 

1.0350  0000 

1.0400  0000 

2 

1.0201  000 

1.0302  250 

1.0404  0000 

1.0506  2500 

1.0609  0000 

1.0713  2500 

1.0816  0000 

3 

1.0303  010 

1.0456  784 

1.0613  0800 

1.0768  9062 

1.0927  2700 

1.1087  1787 

1.1248  6400 

4 

1.0406  040 

1.0613  636 

1.0824  3216 

1.1038  1289 

1.1255  0881 

1.1475  2300 

1.1698  5856 

5 

1.0510  101 

1.0772  840 

1.1040  8080 

1.1314  0821 

1.1593  7407 

1.1876  8631 

1.2166  5290 

« 

1.0615  203 

1.0934  433 

1.1261  6243 

1.1596  9342 

1.1940  5230 

1.3393  5533 

1.3653  1903 

7 

1.0721  354 

1.1098  450 

1.1486  8567 

1.1886  8575 

1.2398  7387 

1.3733  7926 

1.3159  3178 

8 

1.0828  567 

1.1264  926 

1.1716  5938 

1.2184  0290 

1.2667  7008 

1.3168  0904 

1.3685  6905 

9 

1.0936  853 

1.1433  900 

1.1950  9257 

1.2488  6297 

1.3047  7318 

1.3628  9735 

1.4233  1181 

10 

1.1046  221 

1.1605  408 

1.2189  9443 

1.2800  8454 

1.3439  1638 

1.4105  9876 

1.4803  4438 

11 

1.1156  683 

1.1779  489 

1.3433  7431 

1.3120  8666 

1.3843  3387 

1.4599  6972 

1.5394  5406 

12 

1.1268  250 

1.1956  182 

1.3683  4179 

1.3448  8883 

1.4257  6089 

1.5110  6866 

1.6010  3233 

13 

1.1380  933 

1.2135  524 

1.2936  0663 

1.3785  1104 

1.4685  3371 

1.5639  5606 

1.6650  7351 

U 

1.1494  742 

1.2317  557 

1.3194  7876 

1.4129  73;2 

1.5125  8972 

1.6186  9452 

1.7316  7645 

15 

1.1609  690 

1.2503  321 

1.3458  6834 

1.4483  9817 

1.5579  6742 

1.6753  4883 

1.8009  4351 

16 

1.1725  786 

1.2689  855 

1.3727  8570 

1.4845  0568 

1.6047  0644 

1.7339  8601 

1.8729  8135 

17 

1.1843  044 

1.2880  203 

1.4003  4143 

1.5216  1826 

1.6528  4763 

1.7946  7555 

1.9479  0050 

18 

1.1961  475 

1.3073  406 

1.4383  4625 

1.5596  5872 

1.7024  3306 

1.8574  8920 

2.0258  1652 

19 

1.2081  090 

1.3269  507 

1.4568  1117 

1.5986  5019 

1.7535  0605 

1.9225  0132 

2.1068  4918 

20 

1.2201  900 

1.3468  550 

1.4859  4740 

1.6386  1644 

1.8061  1123 

1.9897  8886 

2.1911  2314 

21 

1.2323  919 

1.3670  578 

1.5156  6634 

1.6795  8185 

1.8603  9457 

2.0594  3147 

2.2787  6807 

22 

1.2447  159 

1.3875  637 

1.5459  7967 

1.7215  7140 

1.9161  0341 

2.1315  1158 

2.-3699  1879 

23 

1.2571  630 

1.4083  773 

1.5768  9926 

1.7646  1068 

1.9735  8651 

2.2061  1448 

2.4647  1555 

24 

1.2697  346 

1.4395  028 

1.6084  3725 

1.8087  2595 

2.0327  9411 

3.8833  3849 

2.5633  0417 

25 

1.2824  320 

1.4509  454 

1.6406  0599 

1.8539  4410 

2.0937  7793 

3.3633  4498 

2.6658  3633 

26 

1.2952  563 

1.4727  095 

l.(f?34  1811 

1.9002  9270 

2.1565  9127 

8.4459  5856 

2.7724  6979 

27 

1.3082  089 

1.4948  002 

1.7068  8648 

1.9478  0003 

2.2213  8901 

8.5315  6711 

2.8833  6858 

28 

1.3212  910 

1.5172  222 

1.7410  2421 

1.9964  9503 

3.3879  2768 

2,6201  7196 

2.9987  0333 

29 

1.3345  039 

1.5399  805 

1.7758  4469 

2.0404  0739 

3.3565  6551 

2.7118  7798 

3.1186  6145 

30 

1.3478  490 

1.5630  802 

1.8113  6158 

3.0975  6758 

2.4273  6347 

2.8067  9370 

3.2433  9751 

31 

1.3613  274 

1.5865  264 

1.8475  8882 

3.1500  0677 

3.5000  8035 

2.9050  3148 

3.3731  3341 

32 

1.3749  407 

1.6103  243 

1.8845  4059 

3.2037  5694 

3.5750  8276 

3.0067  0759 

3.5080  5875 

33 

1.3886  901 

1.6344  792 

1.9223  8140 

2.2588  5086 

2.6523  3524 

3.1119  4235 

3.6483  8110 

34 

1.4025  770 

1.6589  964 

1.9606  7603 

2.3153  2313 

2.7319  0530 

3.2208  6033 

3.7943  1634 

35 

1.4166  028 

1.6838  813 

1.9998  8955 

3.3733  0519 

2.8138  6245 

3.3335  9045 

3.9460  8899 

36 

1.4307  688 

1.7091  395 

3.0398  8734 

2.4335  3533 

3.8983  7833 

3.4502  6611 

4.1039  3355 

37 

1.4450  765 

1.7347  766 

3.0806  8509 

2.4933  4870 

3.9853  2668 

3.5710  2543 

4.2680  8986 

38 

1.4595  272 

1.7607  983 

2.1222  9879 

2.5556  8242 

3.0747  8348 

3.0960  1132 

4.4388  1345 

39 

1.4741  225 

1.7872  103 

2.1647  4477 

2.6195  7448 

3.1670  2698 

3.8253  7171 

4.6163  6599 

40 

1.4888  637 

1.8140  184 

2.2080  3966 

3.6850  6384 

3.2620  3779 

3.9592  5973 

4.8010  2063 

41 

1.5037  524 

1.8413  287 

2.2523  0046 

3.7531  9043 

3.3598  9893 

4.0978  3381 

4.9930  6145 

42 

1.5187  899 

1.8688  471 

3.3973  4447 

3.8209  9520 

3.4606  9589 

4.2413  5799 

5.1927  8391 

43 

1.5339  778 

1.8968  798 

3.3431  8936 

2.8915  2008 

3.5645  1677 

4.3897  0302 

5.4004  9527 

44 

1.5493  176 

1.9253  330 

3.3900  5314 

3.9638  0S08 

3.6714  5227 

4.5433  4160 

5.6165  1508 

45 

1.5648  107 

1.9543  130 

3.4378  5431 

3.0379  0328 

3.7815  9584 

4.7023  5855 

5.8411  7568 

46 

1.5804  589  ' 

1.9835  263 

3.4866  1139 

3.1138  5086 

3.8950  4372 

4.8669  4110 

6.0748  2271 

47 

1.5962  634  i 

3.0132  791 

3.5363  4351 

3.1916  9713 

4.0118  9503 

5.0373  8404 

6.3178  1562 

48 

1.6122  261  1 

3.0434  783 

3.5870  7039 

3.2714  89:)6 

4.1323  5188 

5.2135  8898 

6.5705  2824 

49 

1.6283  483 

3.0741  305 

3.6388  1179 

3.3532  7630 

4.2563  1944 

5.3960  6459 

6.8333  4937 

50 

1.6446  318 

2.1052  434 

2.6915  8803 

3.4371  0872 

4.3839  0603 

5.5849  2686 

7.1066  8335 

COMPOUND    INTEREST  TABLE. 


225 


Compound    Interest  Table. 


Showing  the  amount  of  $1  id  compound  interest,  at  various  rates  per  cent, 
for  any  number  of  years,  from  1  year  to  50  years,  inclusive. 


Yrs. 


6 
7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 

18 
19 
20 

21 
22 
23 
24 
25 

26 
27 

28 
29 
30 

31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 
42 
43 
44 
45 

46 
47 

48 
49 
50 


4}4  V^^  ^^' 


1.0450  0000 
1.0920  2500 
1.1411  6612 
1.1925  1860 
1.2461  8194 

1.3022  6012 
1.3608  6183 
1.4221  0061 
1.4860  9514 
1.5529  6942 

1.6228  5305 
1.6958  8143 
1.7721  9610 
1.8519  4492 
1.9352  8244 


7015 
7681 
7877 
6031 
1402 

4116 
5201 
6635 
1383 
3446 


2.0223 
2.1133 
2.2084 
2.3078 
2.4117 

2.5202 
2.6336 
2.7521 
2.8760 
3.0054 

3.1406  7901 

3.2820  0956 

3.4296  9999 

3.5840  3649 

3.7453  1813 

3.9138  5745 
4.0899  8104 
4.2740  3018 
4.4663  6154 
4.6673  4781 

4.8773  7846 
5.0968  6049 
5.3262  1921 
5.5658  9908 
5.8163  6454 

6.0781  0094 
6.3516  1548 
6.6374  3818 
6.9361  2290 
7.2482  4843 

7.5744  1961 
7.9152  6849 
8.2714  5557 
8.6436  7107 
9.0326  3627 

15 


5  per  ct. 


1.0500  000 
1.1025  000 
1.1576  250 
1.2155  063 
1.2762  816 

1.3400  956 
1.4071  004 
1.4774  554 
1.5513  282 
1.6288  946 

1.7103  394 
1.7958  563 
1.8856  491 
1.9799  31G 

2.0789  282 

2.1828  746 
2.2920  183 
2.4066  192 
2.5269  502 
2.6532  977 

2.7859  626 
2.9252  607 
3.0715  238 
3.2250  999 
3.3863  549 

3.5556  727 
3.7334  563 
3.920f  291 
4.1161  35G 
4.3219  424 

4.5380  395 
4.7649  415 
5.0031  885 
5.2533  480 
5.5160  154 

5.7918  161 
6.0814  069 
6.3854  773 
6.7047  512 
7.0399  887 

7.3919  882 
7.7615  876 
8.1496  669 
8.5571  503 
8.9850  078 

9.4342  582 

9.9059  711 

10.4012  697 

10.9213  331 

11.4673  998 


6  per  ct. 


1.0600  000 
1.1236  000 
1.1910  160 
1.2624  770 
1.3382  256 

1.4185  191 
1.5036  303 
1.5938  481 
1.6894  790 
1.7908  477 

1.8982  986 
2.0121  965 
2,1329  283 
2.2609  040 
2.3965  582 

2.5403  517 
2.6927  728 
2.8543  392 
3.0255  995 
3.2071  355 

3.3995  636 
3.6035  374 
3.8197  497 
4.0489  346 
4.2918  707 

4.5493  830 
4.8223  459 
5.1116  867 
5.4183  879 
5.7434  912 

6.0881  006 
6.4533  867 
6.8405  899 
7.2510  253 
7.6860  868 

8.1472  520 
8.6360  871 
9.1542  524 
9.7035  075 
10.2857  179 

10.9028  610 
11.5570  327 
12.2504  546 
12.9854  819 
13.7646  108 

14.5904  875 
15.4659  167 
16.3938  717 
17.3775  040 
18.4201  543 


7  per  ct. 


1.0700  000 
1.1449  000 
1.2250  430 
1.3107  960 
1.4025  517 

1.5007  304 
1.6057  815 
1.7181  862 
1.8384  592 
1.9671  514 

2.1048  520 
2.2521  916 
2.4098  450 
2.5785  342 
2.7590  315 

2.9521  638 
3.1588  152 
3.3799  323 
3.6165  275 
3.8696  845 

4.1405  624 
4.4304  017 
4.7405  299 
5.0723  670 
5.4274  326 

5.8073  529 
6.2138  676 
6.6488  384 
7.1142  571 
7.6122  550 

8.1451  129 
8.7152  708 
9.3253  398 
9.9781  135 
10.6765  815 

11.4239  422 
12.2236  181 
13.0792  714 
13.9948  204 
14.9744  578 

16.0226  699 
17.1442  568 
18.3443  548 
19.6284  596 
21.0024  518 

22.4726  234 
24.0457  070 
25.7289  065 
27.5299  300 
29.4570  251 


8  per  ct. 


1.0800  000 
1.1664  000 
1.2597  120 
1.3604  890 
1.4693  281 

1.5668  743 
1.7138  243 
1.8509  302 
1.9990  046 
2.1589  250 

2.3316  390 
2.5181  701 
2.7196  237 
2.9371  936 
3.1721  691 

3.4259  426 
3.7000  181 
3.9960  195 
4.3157  Oil 
4.6609  571 

5.0338  337 
5.4365  404 
5.8714  637 
6.3411  807 
6.8484  752 

7.3963  532 
7.9880  615 
8.6271  064 
9.3172  749 
10.0626  569 

10.8676  694 
11.7370  830 
12.6760  496 
13.6901  336 
14.7853  443 

15.9681  718 
17.2456  256 
18.6252  756 
20.1152  977 
21.7245  215 

23.4624  832 
25.3394  819 
27.3666  404 
29.5559  717 
31.9204  494 

34.4740  853 
37.2320  122 
40.2105  731 
43.4274  190 
46.9016  125 


9  per  ct, 


1.0900  000 
1.1881  000 
1.2950  290 
1.4115  816 
1.5386  240 

1.6771  001 
1.8280  391 
1.9925  626 
2.1718  933 
2.3673  637 

2.5804  264 
2.8126  648 
3.0658  046 
3.3417  270 
3.6424  825 

3.9703  059 
4.3276  334 
4.7171  204 
5.1416  613 
5.6044  108 

6.1088  077 
6.6586  004 
7.2578  745 
7.9110  832 
8.6230  807 

9.3991  579 
10.2450  821 
11.1671  395 
12.1721  821 
13.2676  785 

14.4617  695 
15.7633  288 
17.1820  284 
18.7284  109 
20.4139  679 

22.2512  250 
24.2538  353 
26.4366  805 
28.8159  817 
31.4094  200 

34.2362  679 
37.3175  320 
40.6761  098 
44.3369  597 
48.3272  861 

52.6767  419 
57.4176  486 
62.5852  370 
68.2179  083 
74.3575  201 


10  per  ct. 


1.1000  000 
1.21CX)  000 
1.3310  000 
1.4641  000 
1.6105  100 

1.7715  610 
1.9487  171 
2.1435  888 
2.3579  477 
2.5937  425 

2.8531  167 
3.1384  284 
3.4522  712 
3.7974  983 

4.1772  482 

4.5949  730 
5.0544  703 
5.5599  173 
6.1159  390 
6.7275  000 

7.4002  499 
8.1402  749 
8.9543  024 
9.8497  327 
10.8347  059 

11.9181  765 
13.1099  942 
14.4209  936 
15.8630  930 
17.4494  023 

19.1943  425 
21.1137  768 
23.2251  544 
25.5476  699 
28.1024  369 

30.9126  805 
34.0039  486 
37.4043  434 
41.1447  778 
45.2592  556 

49.7851  811 
54.7636  993 
60.2400  692 
66.2640  761 
72.8904  8a7 

80.1795  321 

88.1974  853 

97.0172  338 

106.7189  573 

117.3908  529 




226  EXAMPLES  IN"  INTEREST. 

Remarks. — 1.  To  find  the  amount  of  any  given  principal,  for  any  required  number  of  years: 
multiply  the  given  principal  by  the  amount  of  $1  at  the  given  rate,  as  shown  by  the  table. 

2.  For  periods  beyond  the  scope  of  the  table :  multiply  together  the  amounts  shown  for  periods 
the  sum  of  which  will  equal  the  time  required.  For  example,  to  find  the  compound  amount 
of  $1  for  100  years:  multiply  the  amount  for  50  years  by  itself;  to  find  the  compound  amount  for 
75  years:  multiply  the  amount  for  50  by  that  for  25;  of  30  by  45;  of  37  by  38;  of  40  by  35,  etc. 

3.  If  interest  is  to  be  compounded  semi-annually,  take  one-balf  the  rate  for  twice  the  time. 

4.  If  interest  is  to  be  compound  quarterly,  take  one  fourth  the  rate  for  four  times  the  time. 

5.  If  interest  is  to  be  compounded  bimonthly,  take  one-sixth  the  rate  for  six  times  the  time. 

6.  To  find  the  compound  amount  of  print  ipals  of  $100,  or  less:  multiply  the  principal  by 
the  amount  as  shown  in  the  table,  using  only  3  of  its  decimal  places.  For  principals  of  $1000, 
or  less,  use  only  4  of  the  decimal  places,  and  so  on. 

736.  To  find  the  Principal  or  Present  Worth  of  an  Amount  at  Compound 
Interest. 

Example. — "What  principal  will,  in  3  yr.,  at  6^,  amount  to  $5955.08,  if  inter- 
est is  compounded  annually.  ? 

Explanation. — From   the   table 
Opekation.  find  the  amount  of  $1  for  3  years,  at 


15955.08  =  total  unit. 


6  per  cent,  interest,  compounded  an- 
Eually,  to  be  $1.191016;  to  find  the 
$1.191016  =  amt.  of  U  for  the  rate  and  time.  principal  that  will,  at  the  given  rate 
15955.08  -=-  1.191016  =  $5000,  principal.  andtime,  amount  to  $5955.08,  divide 

$5955.08  by  1.191016. 

Rule. — Divide  the  compouiid  amoiont  given  by  the  compound  amount 
of  one  dollar  for  the  time  and  rate  given. 

EXABIPi:.ES  FOR  PRACTICE. 

737.  1.  What  principal  will,  in  8  years,  at  h'^,  amount  to  $4107.26,  if  interest 
is  compounded  semi-annually  ? 

2.  Find  that  principal  which  will,  in  5  years,  at  8,^^  interest,  compounded 
quarterly,  amount  to  $1516.11. 

3.  At  what  rate  of  interest,  compounded  annually,  must  $1750  be  loaned,, 
that  in  7  years  it  may  amount  to  $2381.51  ? 

Remark. — Divide  the  amount  by  the  principal,  carrying  the  quotient  to  six  decimal  places; 
refer  to  the  7  j^cars  line,  or  column,  for  an  amount  corresponding  to  the  quotient.  The  rate 
column  in  which  it  is  found  will  indicate  the  rate  per  cent,  required. 

Jf..  At  wliat  rate  of  interest,  compounded  annually,  must  $2500  be  loaned  for 
12  years,  that  it  may  accumulate  $3795.43  interest  ? 

MISCELLAKEOUS  EXAMPL,E.S  FOR  PRACTICE. 

738.  1.  What  sum  will,  on  Sept.  5,  1889,  discharge  a  debt  of  $550,  dated 
Mar.  19,  1884,  bearing  9;<^  interest,  if  interest  is  compounded  semi-annually,  and 
no  i)ayments  are  made  until  final  settlement? 

2.  If  nothing  was  paid  until  final  settlement,  what  amount  would  pay  a  debt 
of  $1450,  made  July  15,  1885,  and  ])aid  Dec,  3,  1888,  if  interest  is  allowed  at 
the  rate  of  6^,  compounded  quarterly? 


REVIEW    EXAMPLES   IN   INTEREST.  227 

3.  What  amount  will  be  due  Apr.  15,  1891,  on  a  debt  of  $1100,  created  May 
1,  1887,  if  the  interest  thereon  is  at  the  rate  of  10^,  compounded  semi-annually. 

4.  Sept.  19,  18S6,  I  borrowed  15000,  at  ^i,  and  agreed  that  until  settlement 
was  made  I  would  permit  the  compounding  of  the  interest  every  two  months. 
Under  such  conditions,  what  amount  will  be  due  Oct.  25,  1890  ? 

5.  June  29,  1884,  I  borrowed  some  money  at  8^,  interest  to  be  compounded 
quarterly;  January  5,  1890,  I  paid  $1361.82  in  full  settlement.  What  was  the 
sum  borrowed. 

REVIEW  EXAMPLES   IX   INTEREST. 

739.  L  Smith  loaned  $2400,  at  6^  simple  interest,  until  it  amounted  to 
$3000.     For  what  time  was  the  loan  made  ? 

2.  At  what  rate  per  cent,  per  annum  must  $1080  be  loaned  for  7  yr.  3  mo. 
27  da.,  that  it  may  amount  to  $1611.35  ? 

3.  A  man  invested  $16000  in  business,  and  at  the  end  of  3  yr.  3  mo.,  with- 
drew $22880,  which  sum  included  investment  and  gains.  What  yearly  per  cent. 
of  interest  did  his  investment  pay  ? 

Jf..  Find  the  interest  of  that  sum  for  11  yr.  8  da.,  at  10^^,  which  will,  at  the 
given  rate  and  time,  amount  to  $1715.08. 

5.  Sold  an  invoice  of  crockery  on  2  mo.  credit;  the  bill  was  paid  3  mo.  18  da. 
after  the  date  of  purchase,  with  interest,  at  8^,  by  a  check  for  $1963.45.  How 
much  was  the  interest  ? 

6.  A  debt  of  $7150,  dated  Mar.  27,  1885,  and  bearing  6,^  interest,  payable 
quarterly,  was  paid  in  full  July  5,  1888.  If  no  previous  payments  had  been 
made,  how  much  was  due  at  final  settlement  ? 

7.  A  man  having  $21000,  invested  it  in  real-estate,  from  which  he  received  a 
semi-annual  income  of  $787.50.  He  sold  this  property  at  cost  and  invested  the 
proceeds  in  a  business  which  yielded  him  $472.50  quarterly.  How  much  gi'eater 
rate  per  cent,  per  annum  did  he  received  from  the  second  investment  than  from 
the  first. 

8.  In  order  to  engage  in  business,  I  borrowed  $3750  at  6^,  and  kept  it  until 
it  amounted  to  $4571.25.     How  long  did  I  keep  the  money  ? 

9.  A  bond  and  mortgage,  bearing  8j^  interest,  and  dated  May  1,  1880,  was 
settled  in  full  Nov.  16,  1888,  by  the  payment  of  $17685.  For  what  face  amount 
was  the  bond  and  mortgage  given  ? 

10.  In  wliat  time  will  the  interest  at  8,^  be  three-fifths  of  the  principal. 

11.  What  sum  will  be  due  Jan.  18,  1892,  on  a  debt  of  $5100,  dated  Mar.  17, 
1885,  bearing  interest  at  7^  per  annum,  payable  semi-annually,  if  the  first  five 
payments  were  made  when  due  and  no  subseciuent  payments  were  made  ? 

12.  A  building  which  cost  $10500,  rents  for  $87.50  i)cr  month.  What  annual 
rate  of  interest  on  his  investment  does  the  owner  receive,  if  he  pays  yearly  taxes 
amounting  to  $102.50;  insurance,  $21.25;  repairs,  $136.80;  and  janitor s  services, 
$56.95? 

13.  A  merchant  sold  a  stock  of  glassware  on  one  month's  credit;  the  bill 
was  not  paid  until  3  mo.  21  da.  after  it  became  due,  at  which  time  the  seller 
received  a  draft  for  $4716.21  for  the  bill  and  interest  thereon  at  the  rate  of  5^. 
Find  the  selling  price  of  the  goods. 


22H  REVIEW   EXAMPLES   IN    INTEREST. 

J4.  Oct.  12,  1888,  I  purchased  2700  bushels  of  wheat,  m  $1.05  per  bushel, 
and  afterwards  sold  it  at  a  profit  of  Q^,  On  what  date  was  the  wheat  sold,  if 
my  gain  Avas  equivalent  to  10^  interest  on  my  investment  ? 

15.  A  speculator  borrowed  $G2oO,  at  74^  interest,  and  with  tlie  money  bought 
a  note,  the  face  of  which  was  $7500,  maturing  in  nine  montlis  without  interest, 
but  which  was  not  paid  until  two  years  from  the  date  of  its  purchase.  If  the 
note  drew  6^  interest  after  maturity,  did  its  purchaser  gain  or  lose,  and  how 
much  ? 

10.  I  am  offered  a  house  that  rents  for  $27  per  month,  at  sucli  a  price  that, 
after  paying  $67.20  taxes,  and  other  yearly  expenses  amounting  to  $24.85,  my  net 
income  will  be  Siffc  on  my  investment.     What  is  the  price  asked  for  the  house  ? 

17.  Having  three  girls,  Grace,  Mabel,  and  Flora,  aged  respectively  15,  13, 
and  2  years,  I  wisli  to  invest  such  a  sum  for  each  that  she  may  have  $10000 
on  becoming  of  age.  How  mucli  cash  will  be  required  to  secure  a  4^j^  compound 
interest  investment  ? 

18.  I  loaned  a  friend  a  sum  of  money  for  9  months,  at  G^  per  annum,  and 
when  tlic  loan  was  due  he  paid  $851.50  in  cash,  Avhich  Avas  75,'^  of  the  amount 
due  me;  the  remainder  was  paid  6  mo.  15  da.  later,  witli  interest  at  the  rate  of 
10;*.     Find  the  amount  paid  at  final  settlement. 

10.  Deland  owns  a  summer  resort  valued  at  $45000,  for  which  he  receives, 
for  a  season  of  5  months,  $2400  rent  per  month.  The  year's  expenses  for  taxes, 
repairs,  and  insurances,  average  $0375.  If  he  sells  this  property  and  invests 
the  proceeds  in  a  manufacturing  business  paying  quarterly  $1937.50,  how  much 
will  his  rate  of  interest  be  increased  by  the  change? 

20.  Having  purchased  1150  barrels  of  pork,  at  $1G  per  barrel,  on  4  months' 
credit,  the  dealer,  30  days  later,  sold  it  at  $17.50  per  barrel,  receiving  therefor 
a  G  months'  note  without  interest.  When  the  purchase  money  became  due,  he 
discounted  the  note  on  a  basis  of  7^,  and  paid  his  debt.     How  much  was  gained? 

21.  The  day  Ealph  was  6  years  old  his  father  deposited  for  him  in  a  savings 
bank  such  a  sum  of  money  that,  at  4^  interest,  compounded  quarterly,  there  will 
be  $7500  to  his  credit  on  the  day  he  attains  his  majority.  What  sum  was 
dejjosited? 

23.  December  11,  1887,  a  lumber  dealer  borrowed  money  and  bought  shingles 
at  $4.50  per  M.;  Sept.  17,  1888,  he  sold  the  shingles  and  paid  his  debt,  and  8^ 
interest,  amounting  to  $34G2. GO.     How  numy  thousand  shingles  did  he  buy? 

23.  A  jobber  bouglit  GOOO  yd.  of  Axminster  carpet,  at  $2.80  per  yard,  payable 
in  6  months,  and  immediately  sold  it  at  $3.15  per  yard,  giving  a  credit  of  2 
months;  at  the  expiration  of  tl)e  2  months  he  anticipated  the  payment  of  his 
own  paper,  getting  a  discount  off  of  lO'^  per  annum.  How  mucli  did  he  gain 
by  the  transaction? 

2^.  At  the  age  of  25  a  lady  invested  $3000,  at  7^  per  annum.  What  will  be 
her  age  when  the  investment,  with  its  interest  compounded  semi-annually, 
amounts  to  $1G754. 78? 

2o.  Herbert  is  10  and  Theodore  7  years  old.  If  7^  compound  interest  invest- 
ments can  be  secured  by  their  father,  for  what  amounts  must  such  investments 
be  made  in  order  that  at  tlie  age  of  21  tiie  boys  may  each  have  $]2500? 


REVIEW   EXAMPLES   IN"   INTEREST.  229 

26.  If  money  be  worth  7^  compounded  annually,  which  would  be  better, 
and  how  much,  for  a  capitalist  to  loan  $;25OO0  for  11  years  and  6  months,  than 
to  invest  it  in  land  that,  at  the  end  of  the  time  named,  will  sell  for  $55000 
above  all  expenses  for  taxes  ? 

27.  I  loaned  a  bridge  builder  117500  for  7  years,  at  10^  per  annum,  interest 
payable  quarterly,  and  took  a  bond  and  mortgage  to  secure  the  debt  and  its 
interest.  Nothing  having  been  paid  until  the  end  of  the  7  years,  how  mucli  was 
required  in  full  settlement? 

28.  On  the  20th  of  March,  1888,  I  borrowed  $13500,  at  5^  interest;  on  April 
5  I  loaned  $5000  of  tlie  money  until  Dec.  20,  1888,  at  8^;  April  15,  I  purchased 
with  tlie  remainder  a  claim  for  $10000,  due  Aug.  1,  but  which,  not  being  paid 
at  maturity,  was  extended  until  the  $5000  became  due,  at  the  rate  of  6^.  How- 
much  did  I  gain,  both  claims  having  been  paid  on  the  day  the  loan  of  $5000 
became  due? 

29.  Having  bought  a  mill  for  $12000,  I  paid  cash  $4000  on  delivery,  and  gave 
a  bond  and  mortgage  for  8  years  without  interest  to  secure  the  balance;  to 
secure  the  interest,  which  was  to  be  paid  semi-annually,  at  the  rate  of  1!^  per 
annum,  I  gave  sixteen  non-interest  bearing  notes,  without  grace,  for  $280  each, 
one  maturing  at  the  end  of  each  6  months  for  the  8  years.  If  the  four  of  the 
notes  first  maturing  were  paid  when  due,  and  no  other  payment  was  made  until 
the  mortgage  became  due,  how  much  was  required  for  full  settlement? 

30.  Charles  will  be  11  years  old  Dec.  15,  1888,  John  will  be  8  years  old  July 
28,  and  Walter  was  5  years  of  age  April  30.  If,  on  July  1,  a  6^  compound 
interest  investment  be  made  for  each,  so  that  at  the  age  of  21  he  may  have  810000, 
what  amount  of  cash  will  be  required,  the  interest  being  compounded  quarterly? 


330  TRUE  Discouirr. 


TRUE    DISCOUNT. 

74:0.  Discount  is  an  abatement  or  allowance  made  from  the  amount  of  a 
debt,  a  note,  or  other  obligation,  or  a  deduction  from  the  price  of  goods  for 
payment  before  it  is  due. 

741.  The  Present  "Worth  of  a  debt  payable  at  a  future  time  without  interest, 
is  its  value  tioic;  lience,  is  such  a  sum  as,  being  put  at  simple  interest  at  the  legal 
rate,  will  amount  to  tlie  given  debt  when  it  becomes  due. 

74*2.  True  Discount  is  the  difference  between  the  face  of  a  debt  due  at  a 
future  lime  and  its  present  worth. 

Remarks. — 1.  To  find  present  vs>rth,  apply  the  principles  given  in  Interest.  The  debt 
corresponding  to  the  amount;  the  rate  per  cent,  agreed  upon  to  the  rate;  the  time  intervening 
before  the  maturity  of  the  debt,  to  the  time;  and  the  present  worth,  which  is  the  unknown 
term,  is  the  principal. 

2.  When  payments  are  to  be  made  at  different  times,  without  interest,  find  the  present  worth 
of  each  payment  separately,  and  take  their  sum. 

3.  With  debts  bearing  interest,  and  discounted  at  the  same  or  at  a  different  rate  of  interest, 
the  face  of  the  debt  plus  its  interest  as  due  at  maturity  becomes  the  base. 

743.  To  find  the  Present  Worth  of  a  Debt. 

Example. — Find  the  present  worth  and  true  discount  of  a  claim  for  $871.68, 
due  2  yr.  3  mo.  hence,  if  money  is  worth  C<^  per  annum. 

ExPL.VNATiox. — The  amount  of  the  debt 

,^  at  the  end  of  2  yr.  3  mo.  is  |871.68;  and 

Operatiox.  ,        ^  , ,  .     ,        .  „ 

smce  $1  would  m  that  time,  at  6  per  cent. , 

,133  =  int.  on  $1  for  2  yr,  3  mo.  at  6,<.  amount  to  f  1.135,  the  present  worth  must 

$1,135  =  amount  "       "       "  "  be  as  many  times  $1  as  $1,135  is  contained 

$8:i,68  ---  1.135  =  $768,  present  worth.  *^™<^^  »°  $871.68,  or  $768.    If  the  face  of 

*or«i  />o        «>iao        *ir>o  cfu   i.         J-  i.  the  debt  is  $871.68,  and  its  present  worth 

$871.68  —  $768  =  $103,68,  true  discount.  .    „,     ...;„   ,,    '  .■         ,      n  ., 

'  IS  only   $i68,  the   true  discount   will   be 

$871.68  minus  $768,  or  $103.68. 

IXvii^.— Divide  the  aitvount  of  the  debt,  at  its  maturity,  by  one  dollar 
plus  its  interest  for  the  given  time  and  rate,  and  the  quotient  ivill  be  tlie 
present  worth;  subtract  the  present  ivorth  from  the  amount,  and  the 
remainder  irill  be  the  true  discount. 

KXAMPLKS  FOK  PRACTICE, 

744.  1.  What  is  the  present  worth  of  8661.50,  payable  in  3  yr,  9  mo., 
discounting  at  6,^? 

2.  Find  the  present  worth  and  true  discount  of  a  debt  of  $138.50,  due  in  5 
yr.  6  mo.  18  da.,  if  money  is  worth  7'^  per  annum. 

3.  Find  the  i)resent  worth  of  a  del)t  of  $1750,  $1000  of  which  is  due  in  9 
mo,  and  the  remainder  in  15  mo.,  money  being  worth  6^  per  annum 


EXAMPLES   i:^   TRUE    DISCOUNT.  231 

i.  Which  is  greater,  and  how  mucli,  the  interest,  or  the  true  discount  on 
^516,  due  in  1  yr.  8  mo.,  if  money  is  worth  10$^  per  annum? 

5.  Which  is  better,  and  how  much,  to  buy  flour  at  ^6.75  per  barrel  on  6 
months  time,  or  to  pay  $6  cash,  money  being  Avorth  6^? 

6.  AVhen  money  is  worth  5^  per  annum,  which  is  preferable,  to  sell  a  house 
for  $30,000  cash,  or  $31,000  due  in  one  year  ? 

7.  A  farmer  offered  to  sell  a  pair  of  horses  for  $430  cash,  or  for  $475  dne  in 
15  months  without  interest.  If  money  is  worth  8^  per  annum,  how  much  would 
the  buyer  gain  or  lose  by  accepting  the  latter  offer  ? 

8.  If  money  is  worth  6^,  what  cash  offer  will  be  equivalent  to  an  offer  of 
$1546  for  a  bill  of  goods  on  90  days  credit  ? 

9.  An  agent  paid  $840  cash  for  a  traction  engine,  and  after  holding  it  in 
•stock  for  one  year,  sold  it  for  $933.80,  on  eight  months'  credit.  If  money  is 
worth  6^,  what  was  his  actual  gain  ? 

10.  A  stock  of  moquette  carpeting,  bought  at  81.95  per  yard,  on  8  months' 
credit,  was  sold  on  the  date  of  purchase  for  $1.80  per  yard,  cash.  If  money  was 
worth  6^  per  annum,  what  per  cent,  of  gain  or  loss  did  the  seller  realize  ? 

11.  Marian  is  now  fifteen  months  old.  How  much  money  must  be  invested 
for  her,  at  G^  simple  interest,  that  she  may  have  $15000  of  principal  and  interest 
when  she  celebrates  her  eighteenth  birthday? 

12.  A  thresher  is  offered  a  new  machine  for  $480  cash,  $500  on  3  months 
•credit,  or  $535  on  1  3'car's  credit.  Which  offer  is  the  most  advantageous  for 
him,  and  how  much  better  is  it  than  the  next  best,  with  money  worth  7|^? 

13.  After  carrying  a  stock  of  silk  for  4  months,  I  sold  it  at  an  advance  of  30^ 
on  first  cost,  extending  to  the  purchaser  a  credit  of  one  year  without  interest. 
If  money  is  worth  5,'^  per  annum,  what  was  my  per  cent,  of  profit  or  loss  ? 

14'  Having  bought  a  house  for  $5048  cash,  I  at  once  sold  it  for  $7000,  to  be 
paid  in  18  months  without  interest.  If  money  is  worth  8^  per  annum,  did  I 
gain  or  lose,  and  how  much  ? 

15.  Goods  to  the  amount  of  $510  were  sold  on  6  months'  credit.  If  the 
^selling  price  was  $30  less  than  the  goods  cost,  and  money  is  worth  6^  per  annum, 
how  much  was  the  loss  and  the  per  cent,  of  loss  ? 

16.  How  much  must  be  discounted  for  the  present  payment  of  a  debt  of 
$8741.50,  $3000  of  which  is  on  credit  for  5  mouths;  $3000  for  8  months,  and 
the  remainder  for  15  months,  money  being  worth  10,^  per  annum  ? 

17.  What  amount  of  goods,  bought  on  6  months  time  or  5^  off  for  cash, 
must  be  purchased,  in  order  that  they  may  be  sold  for  $4180,  and  net  the  pur- 
chaser lOj^  profit,  he  paying  cash  and  getting  the  agreed  discount  off? 

18.  A  dealer  bought  grain  to  the  amount  of  $3700,  on  4  months'  credit,  and 
immediately  sold  it  at  an  advance  of  10^.  If  from  the  proceeds  of  the  sale  he 
paid  the  present  worth  of  his  debt  at  a  rate  of  discount  of  S^c  per  annum,  how 
much  did  he  gain  ? 

19.  A  merchant  bought  a  bill  of  goods  for  $3150,  on  G  months'  credit,  and 
the  seller  offered  to  discount  the  bill  5^  for  cash.  If  money  is  worth  7^,^  per 
annum,  how  much  would  the  merchant  gain  by  accepting  the  seller's  offer. 


232  EXAMPLES    IN   TRUE   DISCOUNT. 

20.  The  asking  price  of  a  hardware  stock  is  $5460,  on  which  a  trade  discount 
of  25^,  15^,  and  lOf^  is  offered,  and  a  credit  of  90  days  on  the  selling  price.  If" 
money  is  worth  5^'^,  what  should  be  discounted  for  the  payment  of  the  bill  ten 
days  after  its  purchase  ? 

21.  A  merchant  sold  a  bill  of  goods  for  $1800,  payable  without  interest  in 
three  equal  payments,  in  3  months,  6  months,  and  9  months  respectively.  If 
money  is  worth  5^  per  annum,  how  much  cash  would  be  required  for  full  settle- 
ment on  the  date  of  purchase  ? 

22.  A  stationer  bought  a  stock  worth  $768,  at  a  discount  of  25^  on  the 
amount  of  his  bill,  and  ^'fc  on  the  remainder  for  cash  payment.  He  at  once  sold 
the  stock  on  4  months'  time,  at  lOj:^  in  advance  of  the  price  at  which  it  was  billed 
to  him.  How  much  will  the  stationer  gain  if  his  purchaser  discount  his  bill  on 
the  date  of  purchase  by  true  present  worth,  at  the  rate  of  7^  per  annum  ? 

23.  I  sold  my  farm  for  $10,000,  the  terms  being  one-fifth  cash,  and  the 
remainder  in  four  equal  semi-annual  payments,  with  simple  interest  at  5^  on 
each  from  date;  three  months  later  the  purchaser  settled  in  full  by  paying  with 
cash  the  present  worth  of  the  deferred  payments,  on  a  basis  of  lOj^  per  annum 
for  the  use  of  the  money.     How  much  cash  did  I  receive  in  all  ? 

2J^.  What  amount  of  goods,  bought  on  4  months'  time,  lO,'^  off  if  paid  in  1 
month,  hi)  off  if  paid  in  2  months,  must  be  purchased,  in  order  that  they  may 
be  sold  for  $11480,  and  \  the  stock  net  a  profit  of  15^  and  the  remainder  a 
profit  of  20j^  to  the  purchaser,  if  he  cashes  his  purchase  within  1  month  and 
gets  the  agreed  discount  off  ? 


BANK   DISCOUNT.  233 


BANK    DISCOUNT. 

745.  A  Bauk  is  a  corporation  chartered  by  law  for  the  receiving  and  loaning 
of  money,  for  facilitating  its  transmission  from  one  place  to  another  ])y  means 
of  checks,  drafts,  or  bills  of  exchange,  and,  in  case  of  banks  of  issue,  for 
furnishing  a  paper  circulation. 

Remark. — Some  banks  perform  only  a  part  of  the  functions  above  mentioned, 

746.  Negotiable  Paper  commonly  includes  all  orders  and  promises  for  the 
payment  of  money,  the  property  interest  in  which  may  be  negotiated  or  trans- 
ferred by  indorsement  and  delivery,  or  by  either  of  those  acts. 

747.  Bank  Discount  is  a  deduction  from  the  sum  due  upon  a  negotiable 
paper  at  its  maturity,  for  the  cashing  or  buying  of  such  paper  before  it  becomes 
due. 

748.  The  Proceeds  of  a  Note  or  other  negotiable  paper  is  the  part  paid  to 
the  one  discounting  it,  and  is  equal  to  the  face  of  the  note,  less  the  discount. 

Remark. — In  trve  discount,  the  present  worth  is  taken  as  the  principal ;  in  hank  discount 
the  future  worth  is  taken  as  the  principal. 

749.  The  Face  of  a  Note  is  the  sum  for  which  it  is  given. 

750.  The  Discount  may  be  a  fixed  sum,  but  is  usually  the  interest  at  the 
legal  rate,  and  taken  in  advance. 

751.  The  Time  in  bank  discount  is  always  the  number  of  da3's  from  the  date 
of  discounting  to  the  date  of  maturity. 

752.  The  Term  of  Discount  is  tlie  time  the  note  has  to  run  after  being 
discounted. 

Remark. — Bank  discount  is  usually  reckoned  on  a  basis  of  360  days  for  a  year. 

753.  A  Promissory  Note  is  a  written,  or  partly  written  and  partly  printed, 
agreement  to  pay  a  certain  sum  of  money,  either  on  demand  or  at  a  specified  time. 

Remark. — In  general,  notes  discounted  at  banks  do  not  bear  interest.  If  the  note  be  interest- 
bearing,  the  discount  will  be  reckoned  on  and  deducted  from  the  amount  due  at  maturity. 

754.  Days  of  Grace  are  the  three  days  usually  allowed  by  law  for  the 
payment  of  a  note,  after  the  expiration  of  the  time  specified  in  the  note. 

755.  The  Maturity  of  a  note  is  the  expiration  of  the  days  of  grace;  a  note 
is  due  at  maturity. 

Remarks.— 1.  Notes  containing  an  interest  clause  will  bear  interest  from  date  to  maturity, 
unless  other  time  be  specified. 

2.  Non-interest  bearing  notes  become  interest  bearing  if  not  paid  at  maturity. 

3.  The  maturity  of  a  note  or  draft  is  indicated  by  using  a  short  vertical  line,  with  the  date 
on  which  the  note  or  draft  is  nominally  due  on  the  left,  and  the  date  of  maturity  on  the  right; 
thus,  Oct.  21/24. 


234  GENERAL  BEMARK3   ON   COMMERCIAL   PAPER. 

756.  The  Talue  of  a  note  at  its  maturity  is  its  face,  if  it  does  not  bear 
interest;  if  the  note  is  given  with  interest,  its  value  at  maturity  is  the  face  plus 
the  interest  for  the  time  and  grace. 

Re^iarks. — 1.  Grace  is  given  on  all  negotiable  time  paper  unless  "  mthout  grace"  he  speci&ed. 

2.  In  some  States,  as  Minnesota,  Pennsylvania,  and  others,  drafts  drawn  payable  at  sight  are 
entitled  to  days  of  grace,  and  should  be  accepted  in  the  same  form  as  time  drafts;  while  in  such 
States  drafts  payable  on  demand  have  no  days  of  grace,  and  like  the  sight  drafts  of  most  of  the 
States,  are  dishonored  if  not  paid  on  demand.  Other  States,  as  New  Jersey  and  Pennsylvania, 
have  statutory  requirements  as  to  the  phraseology  of  the  note;  as  to  include  the  phrase  "with- 
out defalcation  or  discount,"  etc.     In  such  matters  State  laws  should  be  observed. 

757.  Notes  given  for  months,  have  their  maturity  determined  by  adding  to 
their  date  the  full  months,  regardless  of  the  number  of  days  thereby  included, 
and  also  the  three  days  of  grace. 

758.  Xotes  given  for  days  have  their  maturity  determined  by  counting  on 
from  their  date  the  expressed  time,  plus  three  days  of  grace.  This  is  done 
regardless  of  the  number  of  months  compassed  by  the  days  so  counted. 

759.  In  some  States,  the  bank  custom  is  to  take  discount  for  both  the  day 
of  discount  and  the  day  of  maturity,  which  is  excessive. 

Rejiarks. — 1.  In  general,  the  laws  of  the  different  States  provide  that,  if  a  note  matures  on 
Sunday,  it  shall  be  paid  on  Saturday;  if  Saturday  be  a  legal  holiday,  then  the  note  shall  be  paid 
on  Friday;  but  the  laws  of  different  States  vary,  and  should  be  carefully  studied  and  fully 
observed,  in  order  to  hold  contingent  parties  responsible. 

2.  Notes  maturing  on  a  legal  holiday  must  be  paid  on  the  day  previous,  if  the  legal  holiday 
occurs  on  Monday,  payment  must  be  made  on  the  preceding  Saturday. 

760.  Banks,  in  many  of  the  larger  cities,  loan  money  on  collateral  securities, 
such  as  stocks,  bonds,  warehouse  receipts,  etc.  Sucli  loans,  being  made  payable 
on  demand,  or  on  one  day's  notice,  are  termed  "  call  loans"  or  '*  demand  loans." 
On  such  the  interest  is  usually  paid  at  the  end  of  the  time. 

Remark. — Variations  in  practice  among  banks,  and  at  the  same  bank  with  different  patrons, 
are  very  common  and  subject  to  no  rule  of  law. 


GENERAL  REMARKS  ON  COMMERCIAL  PAPER. 

761.  Conimercial,  or  Negotiable  Paper,  includes  promissory  notes, 
drafts,  or  bills  of  exchange,  checks,  and  bank  bills,  warehouse  receipts,  and 
certain  other  evidences  of  indebtedness;  but  notes  and  time  drafts  are  the  only 
two  kinds  entering  largely  into  the  operations  of  bank  discount. 

762.  If  there  is  no  admixture  of  fraud  in  the  transaction,  any  negotiable 
paper  may  be  bought  and  sold  at  any  price  agreed  upon  by  the  parties,  and  the 
purchaser  thus  have  full  right  of  recovery. 

763.  The  purchaser  of  a  negotiable  pajier  is  protected  in  his  right  of  recovery 
of  its  amount  agamst  all  original  and  contingent  parties  thereto,  if  he  can  show 
three  conditions  : 

1st.     That  he  gave  value  for  the  paper. 


GENERAL   BEMARKS    OX    COMMERCIAL    PAPER.  235 

2d.      That  he  bought  it  before  its  m<aturity. 

3cl.  That  he  did  not,  at  the  time  of  its  purchase,  know  of  the  existence  of  any 
claim  or  condition  affecting  its  validity. 

764.  Indorsements  are  made  on  notes  for  three  purposes: 
1st.     To  secure  their  payment. 

2d.      To  effect  their  transfer. 

3d.      To  make  a  memorandum  of  a  partial  payment. 

765.  Persons  indorsing  for  security  or  transfer  are  liable  for  the  payment  of 
the  paper  indorsed,  unless  the  holder  of  the  paper  fails  to  demand  payment  of  its 
maker  at  maturity,  and,  in  case  of  its  non-payment,  gives  the  indorser  or  indors- 
ers,  within  a  reasonable  time,  notice  of  its  dishonor  by  the  maker. 

766.  If  the  dishonored  paper  be  foreign — /.  e.,  the  parties  to  it  being  of 
different  states  or  countries  —  to  hold  contingent  parties,  a  formal  notarial  pro- 
test, mailed  to  the  indorsers,  is  required  by  the  laws  of  most  States;  but  a  verbal 
or  other  informal  notice  of  dishonor  is  sufficient  if  the  paper  is  domestic. 

767.  No  demand  notice  or  protest  is  necessary  to  hold  the  maker;  he,  being 
a  principal  debtor,  is  only  released  from  his  obligation  by  the  outlawing  of  the 
note,  or  by  his  i)ayment  of  it. 

768.  A  Protest  is  a  written,  or  partly  written  and  partly  printed,  statement, 
made  by  a  notary  public,  giving  legal  notice  to  the  maker  and  indorsers  of  a  note 
of  its  non-payment. 

769.  The  laws  governing  negotiable  paper  are  not  uniform  throughout  the 
United  States,  and  a  careful  observance  of  the  laws  of  all  the  States  wherein  one 
does  business  is  necessary  to  avoid  risks  of  loss. 

770.  It  is  lawful  to  compute  and  take  interest  for  all  three  of  the  days  of 
^race,  althougli  the  debtor  may  thus  lose  the  interest  for  one  or  two  days  by  the 
fact  that  the  note  matures  on  Sunday  or  on  a  legal  holiday. 

771.  Interest  charges  for  time  of  transfer  of  notes  to  distant  places  for 
demand,  and  for  the  return  of  the  remittance  therefor,  is  a  matter  wholly  of 
custom  with  banks,  as  is  also  an  added  charge  or  fee  for  services  in  relation  to 
such  demand  and  remittance. 

772.  Patrons  of  good  standing  at  banks  are  often  given  credit  for  the  face 
of  interest  bearing  notes  discounted. 

773.  AVhen  a  note  is  discounted  at  a  bank,  the  payee  indorses  it,  thus  making 
it  j)ayable  to  the  bank;  both  maker  and  payee  are  then  responsible  to  the  bank 
for  its  payment. 

774.  Indorsements  for  transfer  are  at  the  same  time  indorsements  for  surety, 
unless  made  "without  recourse." 

775.  Negotiable  i)apers  may  be  transferred: 

1st.  By  indorsement  in  full — i.  e.,  by  the  payee  writing  on  the  back  of  the 
note,  in  substance,  as  follows:     "Pay  to  the  order  of  John  Doe,  Richard  Roe" 


236  EXAMPLES   IN    BANK    DISCOIXT. 

( payee ).  lu  which  event  Doe  becomes  the  legal  owner  of  the  note,  and  possesses 
a  right  to  receive  payment  on  it,  or,  in  case  of  its  non-payment  at  maturity,  to 
sue  and  recover  from  the  maker;  and  if  he  follows  tlie  statute  law  of  the  State 
as  to  demand  and  notice,  he  may  recover,  either  jointly  or  severally,  from  either 
the  maker,  or  from  Roe,  the  indorser,  as  such  indorser  is  also  a  surety. 

2d.  By  indorsement  "  in  blank '' — i.  e.,  by  the  payee  writing,  on  the  back  of 
the  note,  simply  his  name.  After  this  is  done,  the  holder  is  presumed  to  be  the 
owner,  and  he  may,  in  case  of  default,  recover  by  suit  from  the  maker;  and  it 
he  observes  the  requirement  of  the  law  of  the  place,  he  may  also  hold  the  indorser, 
as  such  indorser  becomes  a  surety. 

3d.  By  indorsement  "without  recourse'' — i.  e.,  by  the  payee  writing,  on  the 
back  of  the  note,  in  substance,  ''Pay  John  Doe,  or  order,  without  recourse  to 
me,  Richard  Roe''  (payee);  or  by  writing  simply '•  Without  recourse,  Richard 
Roe."  A  note  so  indorsed  is  fully  transferred  from  the  payee,  but  he  rests  under 
no  obligation  as  to  its  payment. 

776.  The  corresponding  terms  of  Bank  Discount  and  Percentage  are  as 
follows : 

The  Face  of  the  note  =  the  Base. 

The  Rate  Per  Cent.  =  the  Rate. 

The  Bank  Discount  =  the  Percentage. 

777.— To  find  the  Discount  and  Proceeds,  the  Face  of  a  Note,  Time,  and  Rate 
Per  Cent,  of  Discount,  being  given. 

Example. — Find  the  bank  discount  and  proceeds  of  a  note  for  1580,  due  in 

63  days,  at  6^. 

ExPL-VNATios. — The  bank  discount  of  a  note  being  its  inter- 

Operation.  ggj  Jqj.  jjjg  {jjjjg  piiig  grace,  and  the  proceeds  being  the  face  of 

$580.00  =:  face.  ^  °^^^  minus  the  bank  discount,  it  is  only  necessary  to  compute 

n  /,(v y      ^      /.o  ^  the  interest  on  the  face  for  ihafull  time  to  obtain  the  discount, 

!^  "  '      and  to  subtract  such  discount  from  the  face  to  find  the  proceeds; 

$573.91  =  proceeds.  thus,   $6.09   being  the  discount,  $580,   minus  $6.09,  equals. 

$573.91,  proceeds. 

YiuXe.— Compute  the  interest  for  the  time  and  rate,  for  the  hank  discount; 
and  subtract  this  bank  discount  from  the  face  of  the  note,  to  find  the  pro- 
ceeds. 

Remakk. — If  the  note  is  on  interest,  find  the  discount  on  the  amount  of  the  note  at  maturity. 

KXAMPLKS   FOK   PKACTICK. 

778.     1.  Find  the  ])ank  discount  and  i)roceeds  of  a  note  for  $750,  due  in  90 
days,  at  5^. 

2.  Find  the  bank  discount  and  ])roceeds  of  a  note  for  $286.50,  due  in  30 
days,  at  7^. 

3.  Find  the  bank  discount  and  proceeds  of  a  note  for  $1325,  due  in  60  days, 
at  \(H. 


EXAMPLES   IN   BANK    DISCOUNT,  237 

4.  What  is  the  discount  on  a  note  for  $1000,  discounted  at  a  bank  for  23 
days,  at  1^  ? 

f^.     What  are  the  proceeds  of  a  90-day  note  for  $1000,  discounted  at  a  bank 

at  m  ? 

6.  I  paid  in  cash  $950  for  an  engine,  and  sold  it  the  same  day  for  $975, 
taking  a  60-day  note,  which  I  discounted  at  a  bank  at  8,^.  What  was  my  gain 
of  loss  ? 

7.  Find  tlio  bank  discount  and  proceeds  of  a  note  for  $1240,  dated  Sept.  3, 
1888,  i)ayable  in  4  months,  with  interest  at  6^,  and  discounted  Nov.  1,  1888,  at 
the  same  rate. 

8.  What  are  the  proceeds  of  a  note  for  $1750,  due  in  63  days,  bearing  interest 
at  10^,  and  discounted  at  a  bank  at  the  same  rate  ? 

9.  Find  the  maturity,  tefm  of  discount,  and  proceeds  of  the  following  note: 

^286.00.  Buffalo,  N.Y.,  Oct.  25,  1888. 

Three  months  after  date,  1  promise  to  pay  to  the  order  of  Smith  &  Bro.,  Two 
Hundred  Eighty-six  Dollars,  at  the  Erie  County  National  Bank. 
Value  received.  THOMAS  BROWN,  JR. 

Discounted  Jan.  1,  1889,  at  6<. 

10.  Find  the  maturity,  term  of  discount,  and  proceeds  of  the  following  note: 
$800.00.  Cleveland,  0.,  Jan.  31,  1888. 

One  month  after  date,  without  grace,  we  promise  to  pay  to  the  order  of  Hale  £ 
Bly,  Eight  Hundred  Dollars,  with  interest  at  5  per  cent. 
Value  received.  HART  &  COLE. 

Discounted  Feb.  10,  1888,  at  10^. 

11.  Find  tlic  maturity,  term  of  discount,  and  proceeds  of  the  following  note: 
$660.90.  Albany,  N.  Y.,  May  5,  1888. 

Ninety  days  after  date,  I  promise  to  pay  to  the  order  of  H.  H.  Douglas,  Six 
Hundred  Sixty  and  ^W-  Dollars,  toith  interest. 
Value  received.  CLAYTON  S.  METERS. 

Discounted  June  1,  1888,  at  5^. 

12.  Find  the  maturity,  term  of  discount,  and  proceeds  of  the  following  note: 
$2^00.00.  St.  Paul,  Minn.,  Aug.  31,  1888. 

Six  months  after  date,  we  promise  to  pay  to  the  order  of  John  W.  Bell,  Two 
Thousand  Four  Hundred  Dollars,  with  interest  at  8 per  cent,  after  one  month. 
Value  received.  OLIVER  d-  JONES. 

Discounted  Sept.  5,  1888,  at  8^. 

Remarks.— 1.  If  discount  be  required  on  a  basis  of  365  days  for  the  year,  compute  the 
■discount  first  on  a  basis  of  360  days,  and  from  the  discount  so  obtained,  subtract  ^  of  itself. 
2.  The  following  three  examples  are  to  be  worked  on  a  discount  basis  of  365  days. 

13.  Paul  Harmon's  ])ank  account  is  overdrawn  $3596.11  ;  he  now  discounts, 
at  Q^  :  a  90-day  iu)te  for  $450  ;  a  60-day  note  for  $1754.81  ;  a  30-day  note  for 
$851.95  ;  a  20-day  note  for  $345.25  ;  a  10-day  note  for  $100;  proceeds  of  all  to 
his  credit  at  the  bank.  What  is  the  condition  of  his  bank  account  after  he 
receives  these  credits  ? 


238  EXAMPLES   IN   BANK    DISCOTXT. 

H.  Swick  &  Sons'  bank  account  is  overdrawn  $11540.19;  they  now  discount, 
at  d'i  :  a  90-davnote  for  l39T5.:il;  a  60-day  note  for  $5514.25;  a  30-day  note  for 
$1546.19;  a  20-day  note  for  $2546.85;  proceeds  of  all  to  their  credit  at  the  bank. 
What  is  the  condition  of  their  bank  account  after  they  receive  credit  as  above? 

15.  Philo  Perkins  &  Co.'s  bank  account  is  overdrawn  $12,916.47  ;  they  now 
discount,  at  6^:  a  90-day  note  for  $2428.40;  a  GO-day  note  for  $6311.25;  a  30-day 
note  for  $1120.50;  a  26-day  note  for  $4500;  a  10-day  note  for  $1550.50;  Pro- 
ceeds of  all  to  their  credit  at  the  bank.  What  is  the  condition  of  their  bank 
account  after  they  receive  the  above  credits  ? 

779.  To  find  the  Face  of  a  Note,  the  Proceeds,  Time,  and  Rate  Per  Cent,  of 
Discount,  being  given. 

Example. — What  must  be  the  face  of  a  note,  payable  in  60  days,  that,  when 
discounted  at  6^<,  the  proceeds  may  be  $573.91  ? 

Operation.  Explanatiox. — If  the  discount  of  $1,  at 

*,i  r,n       ^  J       L       i  d.1  6  per  cent.,  for  63  days,  is  $.0105,  the  pro- 

$1.00  =  face  of  note  of  $1.  *^,     ,  .,    ,  ,,        /         ; ,  ,    '  ,      . 

ceeds  of  f  1  of  the  note  would  be  $1  minus 

•Q^Q5  =  <^'S-  ^^  ^0*^6  of  $1.  QiQ5^  or  I  9895  .  and  if  the  proceeds  of  *1 

$  .9895  =  proceeds  of  note  of  $1.  are  |.9H95,  it  would  require  as  many  dollars 

face  of   note   to  give   $573.91   proceeds  as 

$573.91  -f-  .9895  =  $580,  face  required.      $.9895  are  contained  times  in  $573.91,  or 

$580. 

Rule. — Divide  the  proceeds  of  the  note  by  the  proceeds  of  one  dollar 
for  the  given  rate  and  tiDie. 

Remark. — If  the  note  be  interest-bearing,  find  the  proceeds  of  one  doUar  of  such  note,  and 
proceed  as  above. 

EXAMPLES   FOR   PRACTICE. 

780.  i.  Whut  must  be  the  face  of  a  OO-day  note  that  will  give  $315.04  pro- 
ceeds, when  discounted  at  6^  ? 

S.     What  face  of  a  30-day  note,  discounted  at  7f^,  will  give  $1241.98  proceeds  ? 

3.  Wishinor  to  borrow  $000  of  a  bank,  for  what  sum  must  mv  90-dav  note  be 
drawn,  to  obtain  the  required  amount,  discount  being  at  lO,'^  ? 

4.  Having  bought  goods  to  the  amount  of  $2431.80  cash,  I  gave  my  60-day 
note  in  settlement.  If  discount  be  at  7^^,  what  should  have  been  the  face  of 
the  note  ? 

5.  What  must  be  the  face  of  a  note  dated  Aug.  16,  1888,  and  payable  6 
months  after  date,  that  when  discounted  at  a  bank  Oci.  1,  1888,  at  6^*^,  it  will 
bring  $2100.55  proceeds  ? 

6.  A  note  dated  Sept.  1,  1888,  payable  in  90  days,  with  interest  at  7^*?,  was 
discounted  21  days  afterdate,  at  lOf^.  If  the  proceeds  were  $690.42,  what  must 
have  been  the  face  ? 

7.  You  have  $328.40  to  your  credit  at  the  bank;  you  give  your  check  for 
$936.20,  after  wliich  you  discount  a  30-day  note  for  $425.40,  proceeds  to  your 
crertit  at  the  bank;  you  also  discount  a  90-day  note  made  by  11.  C.  Davis,  pro- 
ceeds to  your  eredit;  you  now  find  yourself  indebted  to  the  bank  $12.37.  If 
discount  be  at  6^  what  must  have  been  the  face  of  the  Davis  note  ? 


PARTIAL   PAYMENTS*  239 


PARTIAL   PAYMENTS. 

781.  A  Partial  Payment  is  a  part  payment  of  the  amount  of  a  note, 
mortgage,  or  other  obligation  existing  at  the  time  such  jmyment  is  made. 

782.  Part  payments,  or  payments,  are  usually  acknowledged,  and  should 
always  be  by  indorsement  on  the  back  of  the  note  or  other  obligation,  but  some- 
times special  receipts  are  given  for  the  sums  paid.  Indorsements  should  give 
date  and  state  amount  paid;  they  are  then  equivalent  to  receipts. 

783.  Partial  payments  may  apply  to  obligations,  either  before  or  after  their 
maturity. 

784.  A  debtor,  his  attorney,  or  other  authorized  agent,  may  make  a  payment 
either  partial  or  in  full  of  any  obligation,  and  such  payment  may  be  received 
and  receipted  for  by  the  creditor,  his  attorney,  authorized  agent,  or  even  by  one 
not  authorized,  if  such  a  person  occupies  his  place  and  is  so  apparently  his  agent 
as  to  deceive  a  debtor  making  a  payment  in  good  faith. 

785.  Various  rules  are  in  use  for  finding  the  balance  due  on  claims  on  which 
partial  payments  have  been  made ;  but  only  the  United  States  Pule  and  the 
Merchants'  Eule  have  more  than  local  application. 

786.  The  United  States  Rule  is  very  generally  used.  It  has  the  sanction 
of  the  law,  being  the  rule  adopted  by  the  Supreme  Court  of  the  United  States, 
and  has  been  adopted  by  most  of  the  States. 

Remarks. — 1.  It  was  held  by  the  Supreme  Court  of  the  United  States,  in  its  decision  adopting 
or  making  the  above-mentioned  rule,  that  the  payment  should  firet  be  applied  to  cancel  the 
interest ;  that  what  is  left,  if  anything,  after  paying  the  interest,  should  be  used  to  diminish 
the  principal.  In  case  the  payment  is  not  large  enough  to  cancel  the  interest,  it  fails  of  its 
object,  and  is  to  be  passed  as  directed  by  the  rule. 

2.  If  at  the  time  of  the  making  of  a  partial  payment  of  a  debt,  the  debtor  renew  his  obliga- 
tion by  taking  up  the  old  note  or  bond,  and  giving  a  nac  one  bearing  interest  for  the  unpaid 
part  cf  his  debt,  no  taint  of  usury  can  be  shown  affecting  the  validity  of  the  new  note,  even 
though  it  may  be  clearly  shown  that  a  payment  credited  was  less  than  the  interest  due  at  the 
time  such  payment  was  made. 

787.  Principles. — 1.  Paxjments  must  he  applied,  first,  to  tlie  discharge 
of  accrued  interest,  and  then  the  remainder,  if  any,  toward  the  discharge  of  the 
principal. 

2.      Only  unpaid  principal  can  draiv  interest. 

788.  The  Merchants'  Rule  is  used  by  most  banks  and  business  houses, 
where  computations  are  on  sliort  time  obligations,  as  such  rule  is  regarded  as 
the  most  convenient  for  business  purposes. 

Remakk. — The  merchants'  rule  is  varied  in  its  use  by  different  creditors,  and  hence  is 
rather  more  an  agreement,  founded  upon  custom  or  otherwise,  between  debtor  and  creditor  as 
to  mode  of  settlement,  than  a  strict  rule  of  law. 


240  EXAMPLES   IN    PARTIAL   PAYMENTS. 

789.  United  States  Rule  for  Partial  Payments. 
Remark. — Settleuieiits  by  this  rule  are  made  as  follows  : 

ExAMPLE.^-A  note,  tlie  face  of  wliieh  was  $3600,  bearing  interest  at  6^,  was 

given  Oct.  17,  1884,  and  settled  Feb.  14,  1889.     Find  the  balance  due,  the 

following  payments  having  been  made:  Mar.  3, 1885,  $600;  Oct.  25, 1886,  $1000; 

Dec.  6,  1888,  $2400. 

Opeuatiox  and  Explanation. 

Kemakk. — Find  the  time  hy  compound  subtraction. 

Face  of  note 13600.00 

Interest  to  date  of  first  payment  (4  mo.  16  da. ) 81.60 

Amount  of  principal  and  interest  at  time  of  first  payment $3681.60 

First  payment  (of  Mar.  3,  1885) 600.00 

Remainder  after  deducting  first  payment $3081. 60 

Interest  to  date  of  second  payment  (1  yr.  7  mo.  22  da. ) 304.05 

Amount  due  at  time  of  eecond  payment — $3385.65 

Second  payment  (of  Oct.  25,  1886) 1000.00 

Remainder  after  deducting  second  payment $2385. 65 

Interest  to  date  of  third  payment  (2  yr.  1  mo.  11  da.) 302.58 

Amount  due  at  time  of  third  payment $2688.23 

Third  payment  (of  Dec.  6,  1888) 2400.00 

Remainder  after  deducting  third  payment. $288.23 

Interest  to  time  of  settlement  (2  mo.  8  da.) 3.27 

Balance  due  at  time  of  settlement  (Feb.  14,  1889) $291.50 

Rule. — Find  the  amount  of  the  principal  to  the  time  ivhen  tlie  pay- 
ment, or  the  sum  of  the  payments,  shall  equal  or  exceed  the  interest  then 
due;  from  this  amount  deduct  the  payment  or  payments  made;  and 
ivith  the  remainder  as  a  new  principal,  proceed  as  before,  to  the  time  of 
settlement. 

EXAMPLES  FOR  PRACTICE. 

790.  1.  On  a  loan  of  $2000,  made  Mar.  19,  1884,  and  bearing  6^  interest, 
payments  were  made  as  follows:  Nov.  1,  1885,  $500  ;  May  3,  1887,  $700  ;  Feb. 
1, 1888,  $1000.     How  much  will  be  required  for  settlement  in  full.  Mar.  2, 1888? 

2.  Oct.  1,  1885,  a  note  for  $1000  was  given,  payable  in  4  years,  with  6^  inter- 
est. A  payment  of  $50  was  made  1  yr.  from  date;  a  payment  of  $250  Avas  made 
1  yr.  6  mo.  from  date;  a  payment  of  $224  was  made  2  yr.  from  date;  a  payment 
of  $20  was  made  2  yr.  8  mo.  from  date  ;  a  payment  of  $110  was  made  2  yr.  10 
mo.  from  date.     IIow  much  remained  due  at  the  maturity  of  the  note  ? 

3.  On  a  claim  for  $3000,  dated  Aug.  12,  1885,  and  bearing  interest  at  7^, 
payments  were  made  as  follows:  Dec.  15,  1885,  $30;  Apr.  1,  1886,  $550;  Jan. 
20,  1887,  $85;  June  12,  1887,  $1651.50.     IIow  much  was  due  May  30,  1888  ? 

Jf.  I  gave  a  mortgage  for  $10000,  May  9,  1881,  bearing  Q<fo  interest,  and 
made  thereon  the  following  payments:  Sept.  19,  1881,  $500;  Jan.  1, 1883,  $500; 
Apr.  25,  1883,  $4000;  Oct.  15,  1885,  |?4000;  May  1,  1888,  $3525.  How  much 
was  due  iit  final  Gettlement,  June  2,  1888  ? 


EXAMPLES   IN    PARTIAL    PAYMENTS.  241 

5.  The  following  note  was  settled  Oct.  13,  1888  ;  a  payment  of  %'lh  having 
been  made  Feb.  15,  1887  ;  one  of  $300,  July  12,  1887  ;  and  one  of  1200,  Apr.  1, 
1888.     If  money  be  worth  8^,  how  much  was  due  at  final  settlement  ? 

^585.60.  Elmira,  N.Y.,  Aug.  1,  1886. 

Six  months  after  date,  I  promise  to  pay  to  James  H.  Kingshiry,  or  order,  Five 
Hundred  Eighty-five  and  jW  Dollars,  value  received. 

SIMEON  G.  FREEMAIf. 

6.  On  a  mortgage  for  15500,  dated  Aug.  13,  1882,  and  bearing  6^  interest, 
the  following  payments  were  made:  Jan.  1,  1883,  $100;  Mar.  2,  1883, 125;  Aug. 
13,  1885,  $2500  ;  Dec.  19,  1887,  $2500  ;  Mar.  1,  1889,  $500.  How  much  was 
required  for  full  settlement.  Mar.  11,  1889  ? 

7.  On  the  following  note  payments  were  endorsed  as  follows:  Nov.  3,  1886, 
$60  ;  Mar.  16,  1887,  $50;  Oct.  1*  1887,  $50;  Dec.  30,  1887,  $1000;  Apr.  1,  1888, 
^625.     How  much  was  due,  if  paid  in  full  May  8,  1888,  money  being  worth  6^  ? 

41600.00,  Daijton,  Ohio,  Apr.  1,  1886. 

Three  years  after  date,  I  promise  to  p)ay  to  the  order  of  Silas  Hopkins,  One 

Tltousand  Six  Hundred  Dollars,  value  received,  ■with  use. 

PETER  S.  BRYANT. 

8.  On  the  following  note  indorsements  were  made  as  follows:  Aug.  1,  1883, 
•$350;  Nov.  3,  1883,  $1000;  Mar.  20,  1885,  $600;  Mar.  31,  1885,  $2500;  Dec.  11, 
1888,  $2000.     What  was  the  balance  due  Jan.  30,  1889  ? 

46500.00.  Chicago,  III,  Mar.  19,  1882. 

On  demand,  tve  promise  to  pay  to  the  order  of  Ames  <&  Adams,  Six  Thousand 
Five  Hundred  Dollars,  ivith  interest  at  6  per  cent. 

Value  received.  HVRB  d-  HOUGHTON. 

791.     Merchants'  Rule  for  Partial  Payments. 

Example. — Find  the  balance  due  Oct.  13,  1888,  on  a  note  for  $1500,  dated 
July  1,  1887,  bearing  %<;i  interest,  and  on  which  the  following  payments  had 
been  made:    Oct.  1,  1887,  $300;  Feb.  12,  1888,  $420;  June  13,  1888,  $700. 

Operation   and  Explanation. 
Remakk. — Find  the  time  by  compound  subtraction. 

Face  of  note,  dated  July  1,  1887 - $1500.00 

Interest  to  date  of  settlement  (1  yr.  3  mo.  12  d.).. 115.50 

Amount  of  note  at  date  of  settlement. $1615.50 

First  payment  (of  Oct.  1, 1887) f. $300.00 

Interest  of  first  i)ayment  to  date  of  settlement  (1  yr.  12  da.)..  18.60 

Second  payment  (of  Feb.  12,  1888)... 420.00 

Interest  on  second  payment  to  date  of  settlement  (8  mo.  1  da. ) . .  16.87 

Third  payment  (of  June  13,  1888) 700.00 

Interest  on  third  payment  to  date  of  settlement  (4  mo.) 14.00 

Total  amount  of  the  payments. $1469.47 

Balance  due $146.03 

16 


242  EXAMPLES    IN    PARTIAL    PAYMENTS. 

Rule. — Find  the  amoiuit  of  the  principal  to  the  tirne  of  settleithent : 
also  find  the  amount  of  each  payment,  from  the  time  it  was  made  to 
the  time  of  settlement;  siibtra,ct  the  sum  of  the  payments  from  the 
amount  of  the  principal  debt;  the  remainder  will  he  the  halance  due. 

Remabk. — This  rule  is  maiuly  used  in  case  of  short  notes  or  business  accounts. 

KXAMPLES  FOK   PRACTICE. 

792.  1.  What  is  the  bahiuee  due,  Apr.  27,  1889,  on  a  note  for  $1050,  dated 
Jan.  24,  1888,  bearing  ''t't  interest,  if  the  following  indorsements  were  made 
thereon  :  July  1,  1888,  $150;  Oct.  15,  1888,  $400;  Jan.  21,  1889,  $300;  Mar. 
27,  1889,  $60. 

2.  Find  the  bahinee  due  at  the  maturity  of  the  following  note,  payments 
having  been  made  as  follows:  Apr.  1,  1888,  $500;  Aug.  25,  1888,  $1250;  Nov.  3^ 
1888,  $240;  Dec.  30,  1888,  $300;  Feb.  1,  1889,  $200. 

$3000.00.  St.  Louis,  Mo.,  Dec.  3,  1887. 

Eighteen  months  after  date,  I  promise  to  pay  to  the  order  of  Ezra  R.  Andrews^ 
Three  Tliotisand  Dollars,  with  interest  at  5 per  cent. 

Value  received.  GEO.  J.  BEATER. 

3.  How  much  was  due  at  the  maturity  of  the  following  note,  payments  hav- 
ing been  made  as  follows  :  Sept.  11,  1888,  $75  ;  Sept.  19,  1888,  $225  ;  Sept.  26, 
1888,  S159;  Oct.  1,  1888,  $155. 

$650.00.  Wichita,  Kan.,  Sept.  6,  1888. 

Thirty  days  after  date,  I  promise  to  pay  to  Gideon  Piatt  &  Co.,  Six  Hundred 
and  Fifty  Dollars,  with  i?iterest  at  10  per  cent. 

Value  received.  BENJ.  F.  COLEMAN. 

4.  Find  the  balance  due  on  the  following  note,  payments  having  been  made 
as  follows:  May  28,  1888,  $255.50;  June  13,  1888,  $168.41;  Aug.  31,  1888, 
$50  ;  Oct.  30,  1888,  $500  ;   Nov.  1,  1888,  $684.25. 

$2150.00.  Denver,  Colo.,  May  1,  1888. 

Six  months  afterdate,  toe  promise  to  pay  to  the  order  of  Wm.  H.  Sanford,  Two 
Thousand  One  Hundred  Fifty  Dollars,  loith  interest  at  8 per  cent. 

Value  received.  MARTIN  F.  RIONET 

RICHARD  M.  PECK. 


EQUATIOIvr   OF   ACCOUNTS,  243 


EQUATION    OF    ACCOUNTS. 

793.  Equation  of  Accounts,  or  Equation  of  Payments  (called  also 
Averaging  Accomits  or  Averaging  Payments),  is  the  process  of  findiug  the  date 
on  v.'hich  a  single  payment  can  be  made  of  two  or  more  debts  falling  due  at 
different  dates,  or  when  the  balance  of  an  account  having  both  debits  and  credits 
can  be  paid  without  loss  of  interest  to  either  party. 

794.  Accounts  having  entries  on  but  one  side,  either  debit  or  credit,  are 
appropriately  called  simple  accounts,  and  the  process  of  equating  such  accounts 
may  be  called  Simple  Equation. 

795.  Accounts  having  both  debit  and  credit  items  may  likewise  be  called 
compound  accounts,  and  the  process  of  equating  sucli  accounts  may  be  called 
Compound  Equation. 

796.  The  Ayerage  Date  of  Payment,  or  Due  Date,  is  the  date  on  which 
such  i)ayment  or  settlement  nuiy  be  equitably  made;  called  also  the  Equated  Time. 

797.  The  Focal  Date  is  any  assumed  date  of  settlement,  with  which  the 
dates  of  the  several  accounts  are  compared  for  the  purpose  of  finding  the  average 
time  or  due  date. 

Remarks. — 1.  Any  date  conceivable  may  be  taken  as  a  focal  date,  and  interest  may  be 
computed  at  any  rate  per  cent.,  and  either  on  a  common  cr  exact  basis,  without  varying  the 
result;  providing  only  that  the  dates  of  all  items  be  compared  with  such  focal  date,  and 
uniformity  in  rate  and  manner  of  computing  interest  be  observed  throughout. 

2.  In  practice  it  is  vastlj^  better  to  observe  a  simple  method,  by  assuming  the  latest  date  in 
the  account  as  a  focal  date,  computing  all  interest  at  6;;  by  the  short  method,  ou  a  360  day  basis. 

3.  The  importance  of  uniformity,  simplicity,  accuracy,  and  rapidity  in  the  equation  of 
payments  and  accounts  is  such  as  to  justify  the  use  and  repetition  of  the  above  suggestions  as  a 

General  Rule.— //^  all  equations,  extend  time  if  credit  or  time  paper 
he  involved;  select  the  latest  date  as  «-  focal  date,  find  actual  time  in 
days,  and  compute  interest  at  6  per  cent.,  on  a  360  day  basis. 

798.  The  Term  of  Credit  is  the  time  to  elapse  before  a  debt  becomes  due; 
if  given  in  days,  it  is  counted  on  from  the  date  of  purchase  or  sale  the  exact 
number  of  days  of  the  term;  if  given  in  months,  it  is  counted  on  the  number  of 
months,  regardless  of  the  number  of  days  thus  included. 

Remarks. — 1.  Book  accounts  bear  legal  interest  after  they  become  due,  and  notes,  even  if 
not  containing  an  interest  clause,  bear  interest  after  maturity. 

2.  The  importance  of  a  thorough  knowledge  of  both  the  theory  and  practice  of  Equation 
of  Accounts,  on  the  part  of  bookkeepers  and  accountants,  can  hardly  be  over-rated,  a.s  much  of 
this  class  of  work  is  to  be  found  in  every  wholesale  and  commission  business. 

799.  The  equity  of  a  settlement  of  an  account  by  equation  rests  in  the  fact 
that,  by  a  review  of  such  account,  one  of  the  parties  owes  the  other  a  balance  to 


"244  EQUATION    OF   ACCOUNTS. 

which  certain  interest  should  be  added  or  from  which  certain  interest  (discount) 
should  be  subtracted. 

800.  To  find  the  Equated  Time,  when  the  Items  are  all  Debits  or  all  Credits 
and  have  no  Terms  of  Credit. 

Example. — When  does  the  (face)  amount  of  the  following  account  become 
due  by  equation? 
Peter  Dunn,  Directioks. — l.  Take  Nov  1  as  the  focal  date. 

^    T,  1  ,     o    /I         \    ^^     n  2.  Find  the  exact  time  in  days  from  the  date  of 

To  Robt.  S.  Cam]>bell,  Dr.         ,  .,      .    ..    ,     ,  ^  . 
*  each  Item  to  the  focal  date. 

^^^°-  3    Compute  interest  at  6  percent.,  360  day  basis, 

Sept.  5.      To  Mdse. ^60  on  each  item  for  its  time. 

''26.        "        "      100  4.  Find  the  total  of  interest 

Qq^     8.       **        *'  •    200  ^-  I^i^'ide  the  total  interest  by  the  interest  on  the 

Xov    1         "        ''  1"^0  ^^^  amount  for  one  day;  the  quotient  will  be  tLe 

1 average  time  in  days. 

$480  6.  Count  back  from  the  focal  date  the  number  of 

days  average  time  thus  found 

Remark.— Compute  interest  by  rules  on  page  217. 

Operation.  Explanation. — Assume  Nov    1  as  a  focal  date, 

1888.             Items.    Time.        Int.  and  reason  as  follows:     If,  on  Nov.  1,  Dunn  pays 

Sept.  5.      $  60  X  57  =  #  .57  Campbell  the  $120  due  on  that  day,  there  will  be  no 

"  26.         100  X  36  =      .60  interest  charged,  because  that  item  was  paid  when  it 

Oct      8          "^00  X  24  =:       80  became  due.     If,  on  Nov.  1,  Dunn  pays  Campbell 

"V    '     1           1  '>n  N/     0 no  ^^^  ^~^^  ^^^^  ^^^  heen  due  since  Oct.  8,  he  should  pay 

— —                   — —  interest  also  for  the  24  days  between  Oct.  8,  when 

$480                  $1.97  that  item  became  due,  and  Nov.  1,  when,  as  we  have 

Int.  on  $480  for  1  day  =  $  .08  assumed,  it  was  paid,  or  he  should  pay,  or  be  charged 

41  Q*-  _i_    OR 04.4  or  25  dav  with,  $.80  interest      If,  on   Nov.  1,  Dunn  pays  the 

.~  ^  ^               11*  $100  due  Sept.  26,  he  should  pay  interest  also  for  the 

the  average  time;    2o  days  back  ^  ^^^.^  ^0^^.^^^  g^pt  og  ^^d  Nov.  1,  or  $ .  60.    If,  on 

from  Nov.  1  is  Oct.  7.  Nov.  1,  Dunn  pays  the  $60  due  Sept.  5,  he  should  pay 

interest  also  on  that  item  from  its  date  to  Nov.  1,  or  for  57  days,  or  $.57  Now,  on  Nov.  1, 
Dunn  owes  Campbell  not  only  the  $480,  the  total  face  amount  of  the  debt,  but  also  $1.97 
interest;  and  if  a  cash  balance  were  reciuired  Nov.  1,  Dunn  would  owe  $481.97. 

But  the  question  was  not,  what  is  the  cash  balance  due  Nov.  1,  but  when  was  the  $480,  the 
face  amount  of  the  account,  due;  that  is,  from  what  date  should  such  face  amount  draw 
interest,  in  order  that  neither  party  gain  or  lose. 

Now  observe  that  we  have  the  principal,  $480,  the  interest  as  found,  $1.97,  and  the  rate  as 
assumed  and  used,  6  per  cent.,  to  find  the  time.  The  interest  on  $480  for  1  day  is  $  08. 
Since  it  takes  the  principal  1  day  to  accumulate  $  .08,  it  must  have  taken  it  as  many  days  to 
accumulate  $1.97 — or  the  account  was  due  as  many  days  back  from  Nov.  1,  the  focal  date — as 
$.08  is  contained  times  in  $1.97,  or  25  days.  Count  back  25  days  from  Nov.  1,  18S8,  and 
obtain  Oct.  7,  1888,  the  equated  date  of  payment,  or  the  date  on  which  Dunn  could  pay  Campbell 
$480,  the  face  of  the  debt,  •without  loss  of  interest  to  either  party. 

Again:  the  same  example  solved,  when  assuming  Sept.  5,  the  earliest  date,  as  a 
focal  date,  or  by  the  discount  method. 

Remark. — Explanations  like  the  following  are  based  upon  a  settlement  of  accounts,  none 
of  which  are  due  at  the  date  of  settlement  or  adjustment,  as  in  case  of  the  giving  of  an  interest 
bearin"-  note  or  bond  for  the  equitable  amount  due,  or  for  anticipating  the  payments  of  debts, 
thus  requiring  a  cash  balance. 


Operation. 

1888. 

Items.    Time.    Disct. 

Sept.  5. 

$  60  X     0  =  $.00 

"    26. 

100x21=     .35 

Oct.     8. 

200  X  33=  1.10 

Nov.    1. 

120  X  57  =  1.14 

$480               $2.59 

EQUATION^   OF   ACCOUNTS.  245 

Explanation. — Assume  the  earliest  date  (Sept.  5) 
as  the  focal  date,  and  reason  as  follows:  If,  on  Sept. 
5,  Dunn  pays  the  $GOdue  on  that  date,  he  will  neither 
have  to  pay  interest  on  it  nor  be  allowed  discount; 
but  if,  on  Sept.  5,  he  pays  the  $100  due  Sept.  2G,  he 
should  be  allowed  discount  oa  that  item  for  the  21 
days  between  Sept.  5  and  Sept.  26,  or  $  .35  discount. 
If,  on  Sept.  5,  ue  pays  the  |200  not  due  until  Oct.  8. 
he  should  be  allowed  discount  on  that  item  for  the  33  days  between  Sept.  5  and  Oct.  8,  or 
$1.10  discount;  and  if,  on  Sept.  5,  he  pays  the  $120  not  due  until  Nov.  1,  he  should  be  allowed 
discount  on  that  item  for  the  57  days  between  Sept.  5  and  Nov.  1,  or  $1.14  discount.  There" 
fore,  assuming  Sept.  5  as  the  date  of  .settlement,  Dunn  does  not  owe  on  that  date  the  face  amount 
of  the  account,  but  such  amount,  $480,  less  the  amount  of  the  above  discounts,  $2.59,  or  really  a 
cash  balance  of  $480,  minus  $2.59,  or  $477.41.  But  the  question  is  not,  what  was  the  cash 
balance  Sept.  5,  but  on  what  date  would  the  payment  of  the  face  amount,  $480,  have  been 
equitable?  We  have  thus  a  condition  similar  to  that  found  in  the  first  operation,  viz.:  the 
principal,  $480,  the  rate,  6  per  cent.,  and  the  discount  (interest)  given,  to  find  the  time;  and, 
as  before,  divide  the  discount  by  the  discount  on  the  principal  for  1  day,  and  the  quotient,  32, 
will  be  the  average  time  in  days.  And  reason,  in  conclusion,  that  from  Sept.  5  Dunn  is 
entitled  to  retain  the  face  amount  of  his  debt,  $480,  for  32  days,  or  until  it  has  accumulated 
$2.59  interest  in  his  hands;  or,  in  other  words,  in  equity,  he  should  pay  such  amount  32  days 
after  Sept.  5,  or  Oct.  7. 

Again:  same  example,  explained  with  an  intermediate  date  (Oct.  1)  assumed 
as  a  focal  date. 

Opebatiok. 

Interest  on    $60  from  Sept.    5  to  Oct.  1,  26  days  =  $.26 

Interest  on  $100  from  Sept.  26  to  Oct.  1,    5  days  =  .0833  + 

Total  interest,         .         .         .         .  $.3433  + 

Discount  CM  $200  from  Oct.  8  back  to  Oct.  1,    7  days  =  $.2333  + 
Discount  on  $120  from  Nov.  1  back  to  Oct.  1,  31  days  =    .62 

Total  discount,         .        .        .        .        $.8533  + 

.8533H .3433+  =  $.51,  excess  of   discount.     $.51  -^' .08  =  6  days. 

Oct.  1  +  6  days  =  Oct.  7. 

Explanation.— Assume  Oct.  1  as  the  focal  date,  and  reason  as  follows;  If,  on  Oct.  1,  Dunn 
pays  the  $60  due  Sept.  5,  he  should  also  pay  interest  on  that  item  for  the  26  days  between  Sept. 
5,  when  it  became  due,  and  Oct.  1,  when  it  was  (assumed  to  have  been)  paid,  or  he  should 
pay  or  be  charged  with  $.20  interest.  If,  on  Oct.  1,  he  pays  the  $100  due  on  Sept.  26,  he 
should  also  pay  interest  ou  that  item  for  the  5  days  between  Sept.  26,  when  it  became  due, 
and  Oct.  1,  when  it  was  (assumed  to  have  been)  paid,  or  he  should  pay  $  .0833-j-  interest;  thus 
we  have  a  total  interest  charge  against  him  of  $  .3433+  on  the  two  items  of  his  account  not 
paid  until  after  they  were  due.  But  if,  on  Oct.  1,  he  pays  the  $200  not  due  until  Oct.  8,  he 
should  be  allowed  a  discount  for  the  7  days  between  Oct.  8,  when  it  became  due.  and  Oct.  1, 
when  it  was  paid,  or  he  should  be  allowed  a  discount  of  $  .23334-  on  that  item;  and  if,  ou  Oct. 
1,  he  pays  the  $120  not  due  until  Nov.  1,  he  should  be  allowed  a  discount  on  that  item  for  the  31 
days  between  Nov.  1,  when  it  became  due,  and  Oct.  1,  when  it  was  paid,  or  he  should  be  allowed 
a  discount  of  $  .62.  Thus  we  have  a  total  discount  to  be  allowed  him  of  $  .8533+  off  from 
the  two  items  of  his  account  which  he  paid  before  they  were  due.  The  ditference  between 
the  amount  of  interest  charged  to  him,  $  .3433+,  and  the  amount  of  discount  for  which  he 
is  given  credit,  $.8533+,  is  $.51,  an  excess  of  discount,  showing  that  at  the  date  assumed 


246  EQUATION  OF  ACCOUNTS. 

(Oct.  1)  he  does  not  owe  the  face  amount  of  the  account,  $480.  but  $480,  the  face  amount,  less 
$  .5]  discount,  or  only  $479.49,  which  .sum  is  the  cash  balance  duo  on  that  date  (Oct.  1).  But 
since,  as  before,  the  question  is  not  as  to  the  cash  balance,  but  is  the  date  on  which  equitable 
settlement  could  have  been  effected  by  the  payment  of  the  face  amount  of  the  account,  $480, 
we  have,  as  before,  the  principal,  rate,  and  discount  (interest)  given,  to  find  the  time.  Divide 
the  discount,  $.51  by  $.08,  and  tind  Dunn  to  be  entitled  to  withhold  or  delay  the  payment 
of  the  $480  until  it  accumulates  $  .ol  interest  (discount)  in  his  hands,  or  that  he  keep  the  $480 
for  6  days  after  Oct.  1,  thereby  in  equity  paying  it  on  Oct.  7,  as  already  twice  shown. 

Remarks. — 1.  The  above  explanation  is  given  in  addition  to  the  former  two,  in  order  to 
illustrate  that  aut/  date  may  be  used  as  a  focal  date,  and  for  the  object  of  aiding  the  teacher  in 
imparting  to  the  pupil  a  full  understanding  of  the  underlying  principles  iifvolved,  and  it  gives 
added  assurance  that  the  solutions  before  given  led  to  a  correct  result.  yelnot»«<'  of  them,  nor 
all  of  them  taken  together,  can  be  accepted  as  being  anything  beyond  aMurances.  Tliey  are 
not  proofs. 

2.  If  settlement  on  Oct.  7  be  equitable,  the  interest  on  such  of  the  accounts  &s  fall  due 
before  that  date  must  be  offset  or  balanced  hy  the  discount  (interest)  of  such  of  the  accounts  as 
fall  due  after  that  date,  to  within  less  than  one-half  of  the  interest  (discount)  of  the  face  amount 
of  the  account  for  one  day;  otherwise  the  due  date  as  determined  would  be  proven  wrong. 

Proof. — Oct.  7  a.s  a  focal  date. 

Explanation. — Assume  Oct.  7  as  a  focal  date, 
and  reason  as  follows.  If,  on  Oct.  7,  Dunn  pays 
the  $60  due  Sept.  5,  he  should  pay  interest  also 
on  that  item  for  the  32  days  between  Sept.  5  and 
Oct.  7,  or  ^-.32  interest;  and  if,  on  Oct.  7,  he  pays 
the  $100  due  Sept.  26,  he  should  pay  interest 
on  that  till  for  the  11  days  between  Sept.  26  and 
Oct.  7,  or  $.1833-|- interest;  being  thus  charged 
$  .5033+  interest  on  the  two  items  not  paid  until 
after  they  were  due.  But  if,  on  Oct.  7,  he  pays  the 
$200  not  due  until  Oct.  8,  he  should  be  allowed  a 
discount  on  that  item  for  the  1  day  between  Oct. 
$  .03  7  and  Oct  8,  or  $  .0333+  discount;  and  if,  on  Oct. 

7,  he  pays  the  $120  not  due  until  Nov.  1,  he  should  be  allowed  a  discount  on  that  item  for  the  25 
days  between  Oct.  7  and  Isov.  1,  or  $.50  discount;  being  thus  allowed  a  total  discount  of 
$  .5333+  for  the  pre-pajTiient  of  the  items  of  the  account  coming  due  after  Oct.  7.  The 
difference  lM?tween  the  amount  of  the  interest  on  the  items  of  the  account  falling  due  before 
Oct.  7,  from  their  rcsixjctive  dates  down  to  Oct.  7,  and  the  amount  of  the  discounts  on  the  items 
of  the  account  coming  due  after  Oct.  7  from  their  respective  dates  back  to  Oct.  7,  is  only  $.03, 
or  is  le.ss  than  one-half  the  interest  (or  discount)  on  the  face  amount  of  the  account  for  one 
day,  thus  proving  Oct.  7  to  be  the  date  on  which  the  payment  of  the  face  amount  of  the 
accoiuit,  $480,  will  effect  an  equitable  .settlement  between  Dunn  and  Campbell. 

Rule.— I.  Select  tlie  latest  date  as  a  focal  date ;  find  the  time  in  days 
from  the  date  of  each  item  of  the  account  to  the  focal  date,  and  compute 
the  interest  on  each  of  the  respective  items  for  its  time  as  found. 

n.  Divide  the  anwunf  or  sum  of  tJie  interest  on  the  items  hij  the  inter- 
est on  the  face  amount  of  the  account  or  items  for  one  day ;  the  quotient 
ivill  he  the  number  of  days  average  time. 

III.  Count  hack  from  the  focal  date  the  nunibcr  of  days  so  found;  tJis 
date  thus  reached  u-ill  he  the  due  date  of  the  face  amount  of  the  account 
or  the  date  on  ichich  such  face  amount  could  he  paid  without  loss  to 
either  party. 


Operation. 

Days  to  Oct.  7. 

Interest. 

Sept.    5,  $  60 33... 

-$.32 

Sept.  26,    100 11... 

.    .1833  + 

.5033  + 

Discount. 

Oct.   8,    $200 1 

$.0333  + 

Kov.  h      120-...25 

.50 

$.5333  + 

.50.33  + 

EXAMPLES    IN    EQUATION    OF    ACCOUNTS. 


247 


Remarks.— 1.  In  finding  the  average  time  of  credit  in  days,  fractions  of  a  day  of  one-half  or 
greater  are  counted  as  a  full  day;  fractions  less  than  one-half  are  rejected. 

2.  In  business,  odd  days,  odd  cents,  and  even  odd  dollars,  are  often  rejected  in  the  interest 
calculations  in  equating  the  time,  it  being  correctly  reasoned  that,  in  the  long  run,  any  losses  or 
gains  thereby  shown  would  fairly  balance;  and  therefore  business  men,  so  settling,  may 
cut  off  as  they  please.  But  for  class-work,  exact  money,  exact  time,  and  interest  computed  to 
fmir  decimal  places,  should  be  required. 

3.  Any  date  between  the  extremes,  or  within  the  account,  may  be  taken  as  a  focal  date,  the 
only  question  involved  being  a  balance  of  the  interest  or  discount;  but,  except  for  illustrative 
purposes  by  the  teacher,  or  test  exercises  for  advanced  pupils,  the  selection  of  any  date  except 
the  latest  for  a  focal  date  is  not  recommended. 

4.  The  selection  of  the  latest  date  saves  one  interest  computation,  and  removes  the  objection 
often  raised  in  case  an  earlier  or  the  earliest  date  be  chosen,  that  an  account  is  not  likely  to 
have  been  settled  before  it  teas  made. 

5.  The  product  method  of  equating  accounts,  often  used,  and  in  many  cases  capable  of 
producing  correct  results,  is  not  recommended,  because: 

First.  It  is  much  more  difficult  to  comprehend  than  the  interest  method. 

Second.  It  usually  involves  a  greater  number  of  ligures. 

Third.  By  it,  a  cash  balance,  often  desirable,  is  only  obtainable  by  an  additional  operation, 
and  with  difficulty  and  perplexity. 

Fourth.  Equation  of  accounts  having  debit  and  credit  items  is  impossible  by  that  method, 
in  case,  as  frequently  happens,  the  face  amounts  of  the  two  sides  chancfi  to  be  equal;  i.  e.,  the 
debtor  having  paid  the  face  amount  of  his  obligation;  while  there  may  still  be  an  important 
balance  of  interest  or  discount,  which  can  be  readily  adjusted  if  the  interest  method  be  used. 

Fifth.  A  book-keeper,  equating  by  the  interest  method,  can  readily  exhibit  to  his  employer 
the  equity  conditions  of  an  excess  of  interest  or  discount,  even  tliough  the  employer  be  unfa- 
miliar with  the  formal  work  of  the  equation. 

ScGOESTiON  TO  THE  Teacher.  — Placc  On  the  blackboard,  as  an  example,  an  account  with 
a  dozen  or  more  items,  having  different  dates,  and  each  for  a  simple  amount,  and  so  a.ssign  the 
example  that  each  pupil  may  have  a  different  focal  date  from  which  to  work;  then  require 
each  pupil  to  prove  his  result  and  withhold  the  announcement  until  called  for.  Such  exercises 
will  stimulate  the  pupils  to  accuracy  and  speed  in  their  work,  and  will  result  in  imparting  a 
very  thorough  knowledge  of  the  subject. 


EXAMPLES   FOR   PRACTICE. 

801.     When  iire  the  following  uccoimts  due  by  equation: 
Remark. — The  teacher  should  require  that  each  result  be  proved. 


1. 

1888. 

Oct. 


Oct. 


Warren  Pease, 

To  Calvin  Gray,  Ur. 

1,  ToMdse. .- $  ;5 

6,     "      "      50 

14,     "      "      80 

25,     "      *'      120 

31,     "      "      40 

Xorman  Colby, 

To  Seth  Stevens  &  Sons,  Dr. 

1,  To  Mdse 4300 

5,    "        "     - 150 

11,    "        "     120 

IG,    ''        " 200 

28,    "        ''     100 

30,    "        *'     180 


1888. 

Aug. 


Sept.  30, 
Oct.    12, 


Parker  II.  Goodwin,. 

7o  Perkins  &  Ilawley,  Dr. 

7,  To  Mdse. .$200.00 

"      180.55 


Dec. 


3, 


35.60 

100.00 

50.25 


Jan. 

6, 

Feb. 

1, 

a 

27, 

Apr 

3, 

( i 

20, 

<c 

27, 

Wm.  P.  Dugan, 

To  Godfrey,  Son  &  Co.,  Dr. 

To  Mdse. $:iOO 

"      .-   100 

"    :  100 

''      300 

'•      300 

"      200 


248 


EQUATION'    OF    ACCOUNTS. 


5. 

Oct. 

Nov. 

it 

Dec. 

1889. 

Jun. 


1887. 

Nov. 
Dec. 


Jan. 

Feb. 
Mar. 


Gerald,  Joues  &  Co., 

To  Samuel  Smith,  JJr. 

To  Mdse $500.00 

"     821.75 

" 150.00 

" 205.25 

''        "     33.00 


13, 

1, 
28, 
17, 
30, 


300.00 


■i^j  

Theodore  Stanley, 

To  Paul  Fleming,  Dr. 

6,  To  Mdse $500 

28,     ••       •••     - 200 

17,     "       "     ...150 

29'     "       "     .- .150 


13, 
30, 

11, 
31. 


300 
100 
200 
200 


1888. 

Dec. 

<< 
1889. 

Jan. 
Mar. 


Felix  Peterson  &  Bro., 

To  Paul  Paulson  &  Co.,  Dr. 

1,  To  Mdse $1500 

16,     ••        "     2000 


19, 

1, 
21, 


7000 

500 

1000 


S. 

1887. 


Philip  Darling, 

To  Jacob  V.  Hall,  Dr. 
Oct.    6,  To  Mdse. $300 


"  31, 
Nov.  17, 
Dec.    1, 

1888. 

Jan.  20, 
Feb.  16, 
Mar.  3, 
Apr.    6, 


150 

150 
450 

300 
600 
300 
300 


802.     To  find  the  Equated  Time,  when  the  Items  have  Different  Dates,  and 
the  Same  or  Different  Terms  of  Credit. 

Example  (requiring  time    extension). — When  does  the  face  amount  of  the 
following  account  become  due  by  equation? 

John  Price. 

1888.  To  Volney  Clark,  Dr. 

Sept.  14,  To  Mdse.,  1  mo $1000 

"     30,    "        "       5  mo 500 

Nov.  10,    "       "     60da 700 

"     29,    ''        "     30  da 200 

Dec.   31,    "       '*       2  mo 600 

If  the  time  for  the  payment  of  each  of  the  several  items  of  the  above  account 
be  extended  for  the  term  of  credit  indicated,  the  account  will  stand  as  follows: 

John  Price, 

DiRECTioss. — 1.  Assume  the  latest  date  as  a  focal 
date. 

2.  Star  the  focal  date  to  distinguish  it. 

3.  Observe  general  directions  for  example  on  page 
244 


To  Volney  Clark,  Dr. 

1888,  Oct.  14.. $1000 

1889,  Feb.  28. 500 

1889,  Jan.  9 TOO 

1888,  Dec.  29 200 

1889.  Feb.  28 600 


Rule. — I.  Extend  tJie  time  of  credit  of  such  items  as  are  sold  on  credit. 

n.  Select  the  latest  date  as  a  focal  date,  and  find  the  interest  on  each 
item  from  its  maturity  date  to  the  focal  date. 

Ill-  Diiride  the  aggregate  of  interest  thus  found  hy  the  interest  on  the 
fa^e  amount  of  the  account  for  one  day ;  the  quotient  uill  he  the  time  in 
days  to  he  counted  back  from  the  focal  date  to  determine  the  due  date  or 
average  date. 


EXAMPLES    IN    EQUATION    OF    ACCOUNTS. 
EXASTPLKS   FOR   PKACTICK. 


249 


803.  On  what  dates  are  the  face  amounts  of  the  following  accounts  due  Ijy 
equation  ? 

Remarks.— 1.  Extend  the  time,  by  adding  the  term  of  credit  to  the  date  of  each  item,  before 
proceeding  with  the  work. 

2.  Should  two  or  more  items  mature  on  the  same  date,  their  sum  may  be  found,  and  one 
computation  of  interest  serve  for  all. 


1.     Herbert  G.  Williams, 
1888.      '^0  Brewster  &  Brewster,  Dr 
Aug.   15.  To  Mdse.,  2  mo., 

"     29,    "       "  "        300 

Sept.  20,    "       "  "       -  200 

Oct.      4,    "       "  ''        120 

Nov.     I,    "       "  "       100 


2.     Samuel  S.  Sloan, 

1888.  To  A.  D.  Wilton,  Dr. 

Sept.  12,  To  Mdse.,  1  mo., $1000 

"     30,  '*        '*       5  mo., 500 

Nov.  10,  "        ''      60  da., 700 

"     29,  "        "      30  da., 200 

Dec.  31,  "       '*      2  mo., 300 


3.     H.  C.  Colvin, 
1888.                    To  Jas.  Fowler,  Dr. 
Nov.  3,  To  Mdse.,  30  da., $550 

*'  23,    "       "  "       800 

Dec.  1,    "       "  "        90 

**  28,    "       "  "       210 

1889. 

Jan.  11,  "       "  "       600 

*'    31,  "       "  "       300 


o. 

It 
Jan. 
Feb. 
Mar. 
June  29, 


T.  L.  King  &  Son, 
^.  To  Groves  «&  Co.,  Dr. 

30,  To  Mdse.,  1  mo., $  300 

28,     "        ''       60  da., 300 

25,     ''        "      2  mo., 1200 


30  da., 1500 


f^K     John  Jennings, 


1889. 


To  Eichard  Smith,  Dr. 


Jan.    17,  To  Mdse., ..$  50 


-      31, 

a 

a 

1  mo. , 

100 

Feb.      9, 

a 

(< 

2  mo 

600 

Mar.     3, 

ie 

a 

. 

....   200 

June  20, 

a 

a 

3  mo. , 

....   120 

July     8, 

'•■ 

a 

1  mo., 

300 

7.     Porter  Cass  &  Sons, 

1888.  To  Phelps  Bros.,  Dr, 


Feb. 

19, 

To  Mdse. 

,  60  da.. 

..$519.22 

i  i 

29, 

a 

60    " 

..   211.50 

Mar. 

u. 

li 

30    " 

-.   120.00 

a 

25, 

" 

30    " 

-.   181.75 

May 

1, 

a 

2  mo.. 

..     80.00 

a 

31, 

a 

1     '' 

..     69.78 

June 

24, 

a 

3     " 

--  127.75 

^.     01 

1888. 

Oct.  3, 
"     31, 

Dec.  1, 
"    31, 

1889. 

Feb.     3, 

"     28, 

Mar.  12, 

Apr.  30, 


iver  H.  Brown, 
To  Stephen  Brackett,  Dr. 

To  Mdse.,  30  da.,...$  319.50 
"       "       4mo.,  ...     750.00 

"       "        280.50 

"       "        2  mo.,  ...     400.00 

'•'  "  60  da.,...  250.50 

"  " 216.75 

"  " 80.25 

"  "  1  mo.,...  150.00 


<V.     H.  B.  Spencer  &  Co., 

1888.          To  Wood,  Son  &  Co.,  Dr 
Sept.  14,  To  Mdse.,  1  mo., $  1000 

"     30      " 
Nov.  10,'    '' 

"     29,     '-' 
Dec.  31,     '' 

1889. 

Jan.    30,  " 

Feb.   28,  '*' 

Mar.   25,  '' 

June  29,  " 


5  mo., .. 
60  da.,.. 
30  da.,.. 
2  mo.,.. 

500 

70O 

..       200 

600 

1     "    .. 
60  da.,.. 

..       300 

300 

T200 

1500 

250  KQIATIOK    OF    ACCOUNTS. 

804.    To  find  the  Equated  Time,  when  an  Account  has  both  Debits  and  Credits. 

Example. — AVliut  is  the  balance  of  the  following  account,  and  when  due  by 
equation. 
Dr.  James  B.  Greene.  Or. 


1889. 

1889. 

Jan. 

15 

To  Mdse., 

GOO 

Feb. 

1 

By  Cash, 

Feb. 

128 

le         it 

300 

Mar. 

31 

a         ii 

300 
300, 

Directions. — 1.  Select  the  latest  date  as  a  focal  date. 

2.  Find  the  time  from  the  date  (maturity)  of  each  item  to  the  frx-al  date. 

3.  Compute  tbe  interest  on  each  item  for  its  time. 

4.  By  addition,  determine  the  sum  of  the  interest  on  each  side. 

5.  Find  the  difference  between  the  Dr.  and  Cr.  interest  for  an  interest  balance. 

6.  Divide  this  interest  balance  by  the  interest  on  the  balance  of  the  account  for  one  day. 

Operation. 
Dr. 
1889.  Jan.  15,  $600.      T5  days  to  focal  date  =  $7.50,  interest. 
"      Feb.  28,    300.     81     ''     "     "         "   =    1.55, 

Total  Dr.,   ^000.  ^9.05,  total  Dr.  interest. 

Cr. 
1889.   Feb.     1,  %300.     58  days  to  focal  date  =  *2.90,  interest. 

"       *Mar.  3l'.    300.       0     "     ''     "         '"'    == 00, 

Total  Cr.,  $000.  $2.90,  total  Cr.  interest. 

Dr.  balance,  $300. 
Interest  on  $300  for  1  day  =  $.05. 
$9.05  —  $2.90  =  $P).15,  excess  Dr.  interest. 
$6.15  -f-  $  .05  =  123,  or  123  days  equated  time. 
123  days  lack  from  Mar.  31,  1889,  gives  Nov.  28,  1888. 


*  Focal  date. 


Explanation. — Assume  the  latest  date,  Mar.  31,  as  a.  focal  date,  and  reason  as  follows:  If, 
on  Mar.  31,  Greene  receives  credit  for  the  $300  paid  on  that  day,  he  should  not  receive  credit 
for  any  interest,  because  the  money  was  paid  on  the  day  it  fell  due  ;  but  if,  on  Mar.  31,  he 
receives  credit  for  the  $300  that  he  paid  Feb.  1,  he  should  receive  credit  also  for  the  interest 
on  that  payment  for  the  58  days  between  Feb.  1,  when  he  paid  it,  and  Mar.  31,  when  he 
received  credit  for  it,  or  he  should  be  credited  for  $2.90  interest ;  and  if  there  were  no  debits 
or  charges  against  him,  he  would  be  entitled.  Mar.  31, 1889,  to  a  net  credit  of  $602.90,  as  a  cash 
balance  in  his  favor.  But  we  have  the  debit  of  the  account  to  be  considered,  as  follows  :  If, 
on  Mar.  31,  Greene  be  charged  with  $300,  the  value  of  Mdse.  sold  to  him  Feb.  28,  he  should 
also  be  charged  with  its  interest  for  the  31  days  between  Feb.  28  and  Mar.  31,  because  he  did 
not  pay  for  the  Mdse.  when  the  amount  of  it  was  due;  or  he  should,  on  this  item,  be  charged 
$1..55  interest  ;  and  if,  on  Mar.  31,  he  be  charged  with  $600,  the  value  of  Mdse.  sold  him  Jan. 
\~i,  he  should  also  be  charged  with  its  interest  for  the  75  days  between  .Jan.  15  and  Mar.  31, 
because  he  did  not  pay  for  the  Md.se.  when  the  amount  of  it  was  due;  or  he  should,  on  this 
item,  be  charged  $7.50  interest,  thus  being  charged  a  total  of  $9.05  interest,  and  showing  his 
total  debt  to  be  $909.05  on  Mar.  31,  in  case  he  had  received  no  credit  for  payments  made. 
But  since  he  had  received  credit  for  payments  amounting  to  $600,  and  for  interest  thereon 
amounting  to  $2.90,  his  debt,  on  Mar.  31,  was  not  $900,  as  the  sum  of  the  items  charged,  plus 


EQUATION"    OF    ACCOUXTS.  251 

$9.05,  the  sum  of  the  interest  charged,  but  was  $900,  the  sum  charged,  less  $G00,  the  sum 
credited,  or  only  $300  of  principal  debt  or  charge  unpaid,  and  $9.05,  less  $2.90,  or  $6.15, 
interest  balance  due.  And  if  the  cash  balance  due  was  required,  it  would  thus  be  found  to  be 
$306.15.  But  the  question  is  not  concerning  the  cash  balance  due  Mar.  31,  1889,  but  on  what 
date  was  the  $300  balance  of  account  due  by  equation  ?  And  to  determine  this,  proceed  as 
in  the  earlier  explanation  of  this  subject;  having  given  the  principal  (balance  of  account), 
$300,  interest  (balance),  $6.15,  and  rate,  to  find  the  time.  Divide  the  balance  of  interest  by 
the  interest  on  the  balance  for  1  day,  and  find  the  time  to  be  123  days,  and  reason  in  conclusion 
that,  since  on  Mar.  31,  Greene  owed  not  only  the  $300,  but  al.so  $6.15  interest,  he  had  at  that 
date  been  owing  the  $300  for  a  time  sufficient  to  enable  it  to  accumulate  $6.15  interest,  or  for 
123  days;  and  if  he  had,  on  Mar.  31,  1889,  been  owing  the  $300  for  123  days,  that  debt  must 
have  been  due  by  equation  123  days  prior  to  Mar.  31,  1889,  or  since  Nov.  28,  1888. 

For  reference,  and  to  give  assurance  of  the  correctness  of  the  above  conelu.sion, 
tlie  same  example  is  taken  and  solved  with  the  earliest  date  assumed  as  a  focal 

date. 

Operation. 

Dr. 
1889.   *  Jan.  15,  $600.       0  days  to  focal  date  =       00,  discount. 

Feb.  28,    300.     44     "     '•     "         "    =  $3.20,  total  Dr.  discount. 

Total  Dr.,  $900. 

Cr. 
1889.   Feb.     1,  $300.     17  days  to  focal  date  =  $  .85,  discount. 
'•-       Mar.  31,    300.     75     "     "     "         "'    =    3.75, 

Total  Cr.,  $000.  $4.60,  total  Cr.  discount. 

2.20,  total  Dr.  discount. 
Dr.  balance,  $300.  $2.40,  excess  Cr.  discount. 

Interest  or  discount  on  $300  for  1  day,  =  .05. 
$2.40  -^  $.05  =  48  =  number  of  days  equated  time. 
48  days  hack  from  Jan.  15,  1889,  gives  Nov.  S8,  1888. 


*  Focul  datf . 

Explanation.— Assume  Jan.  15,  1889,  the  earliest  date,  as  a  focal  date,  and  reason  as 
follows:  If,  on  Jan.  15,  Greene  pays  the  $600,  the  value  of  Mdse.  bought  on  that  day,  he 
pays  his  debt  when  due,  and  should  neither  be  charged  with  interest  nor  credited  with 
discount;  but  if ,  on  Jan.  15,  he  pays  the  $300  not  due  until  Feb.  28,  he  should  be  credited 
with  discount  on  that  item  for  the  44  days  between  Jan.  15,  when  he  paid  it,  and  Feb.  28, 
when  it  becomes  due  ;  or  he  should  be  credited  with  $2.20  discount  for  the  pre-payment  of  this 
item.  Thus  we  find  that,  on  Jan.  15,  he  did  not  owe  the  $900,  the  face  amount  of  his  debt, 
but  only  $900,  the  face,  less  $2.20  discount.  If  there  were  no  credits  to  be  considered,  he 
would,  on  Jan.  15,  1889,  owe  $897.80  as  a  cash  balance.  But  we  have  to  consider  the  Cr.  of 
his  account,  and  do  so  as  follows:  If,  on  Jan.  15,  he  be  credited  for  the  $300  not  paid  until 
Feb.  1,  he  should  be  charged  discount  on  that  sum  for  the  17  days  between  Jan.  15,  when  he 
received  credit  for  its  payment,  and  Feb.  1,  when  such  payment  was  actually  made,  or 
he  should  be  charged  discount  on  this  item  of  $.85;  and  if,  on  Jan.  15,  he  receives  credit 
for  the  $300,  the  payment  not  made  until  Mar.  31,  he  should  be  charged  discount  on  this 
item  for  the  75  days  between  Jan.  15,  when  he  received  credit  for  its  payment,  and  Mac. 
31,  when  it  was  actually  paid,  or  he  should  be  charged  discount  on  this  item  of  $3.75;  thus 
we  find  that,  on  Jan.  15,  he  should  have  received  credit  for  the  sum  of  his  payments,  $600, 
less  the  sum  of  the  discount,  $4.00,  cliarged  against  him,  or  for  $595.40  as  a  cash  balance;  or 


252  EQUATION   OF  ACCOUNTS. 

that,  on  Jan.  15,  lie  owed  $300  and  stood  charged  with  discount  balance  of  the  difference 
between  $4.60  and  $2.20,  or  $2.40  ;  in  other  words  that,  on  Jan.  15,  1889,  he  not  only  owed 
the  $300,  balance  of  items,  but  also  the  $2.40  balance  of  discount,  or  had  beea  owing  the  $300 
for  a  length  of  time  sufficient  to  enable  that  sum  to  accumulate  $2.40  in  the  creditor's  hand». 
We  have  thus,  as  before  found,  the  principal,  interest  (discount),  and  rate  given  to  find  the 
time;  and  divide  the  interest  (discount)  balance,  $2.40,  by  the  discount  on  the  balance  of  the 
account  for  1  day,  and  find  that,  on  Jan.  15, 1889,  Greene  had  been  owing  the  $300  for  48  days. 
Counting  back  48  days  from  Jan.  15,  1889,  find,  as  before,  the  balance,  $300,  to  have  beeo  due 
by  equation  Nov.  28,  1888. 

Remarks. — 1.  While  the  result,  being  the  same  in  both  the  foregoing  operations,  gives 
assurance  of  the  correctness  of  both,  it  is  assurance  only,  it  is  not  proof. 

2.  If  the  conclusions  drawn  from  the  above  explanations  be  correct,  and  the  balance  be  due 
Nov.  28,  1888,  as  found,  then  the  sum  of  the  discount  of  the  Dr.  items  from  their  respective 
dates  back  to  Nov.  28,  1888,  must  be  offset  or  balanced  by  the  sum  of  the  discount  of  the  Cr, 
items  from  their  respective  dates  back  to  Nov.  28,  1888,  to  within  less  than  one-half  of  the 
discount  of  the  balance.  $300,  for  1  day,  or  to  within  less  than  $.02^. 

Proof, — Take  the  example  as  above  explained,  and  assume  Nov.  28,  1888,  as 

a  focal  date. 

Operation. 

Dr. 
1889.  Jan.  15,  $fiOO.       48  days  back  to  focal  date  =  $4.80,  discount. 
"       Feb.  28,    300.       92     "       "      "     "       ''     -    4.00, 

$9.40,  total  Dr.  discount, 
Cr. 
1889.  Feb.     1,  $300.       65  days  back  to  focal  date  =  $3.25,  discount. 
"      Mar.  31,    300.     123     ''       "     "     "        "     =    6.15, 

$9.40,  total  Cr.  discount. 
Focal  date,  Nov.  28,  1888. 

Explanation. — Assume  Nov.  28,  1888,  as  a  focal  date,  and  compute  the  discount  on  each 
item  of  the  account  for  the  time  between  the  date  of  such  item  and  the  focal  date,  and  find 
that  the  total  of  the  Dr.  discount  exactly  balances  the  total  of  the  Cr.  discount.  Hence  it  is 
proved  that  the  balance  of  the  account  considered  was  due  by  equation  Nov.  28,  1888,  as 
twice  shown. 

Remarks.— 1.  In  case  a  cash  balance  at  any  given  date  is  required,  it  may  be  ascertained 
either  by  computing  the  interest  and  finding  the  amount  on  all  Dr.  items  for  a  total  Dr.,  and 
of  all  Cr.  items  for  a  total  Cr.,  and  by  subtraction  determining  the  balance.  Or,  the  cash 
balance  may  be  ascertained  by  first  finding  the  date  on  which  the  J^ice  balance  of  the  account 
is  due  by  equation,  and  then  adding  interest  in  case  the  due  date  comes  before  the  date  of 
actual  settlement,  or  subtracting  discount  in  case  the  due  date  comes  after  the  date  of  actual 
gettlement. 

2.  After  the  due  date  is  determined,  the  rate  of  interest  or  discount  allowed  should  be 
determined  by  the  law  of  the  place,  or  may  be  by  agreement  of  the  parties;  but  local  interest 
and  usury  laws  would  prevail  in  di.sputed  eases. 

3.  In  proving  the  equation  of  accounts,  the  equitable  settlement  of  which  is  found  to  come 
at  a  date  within  the  account  or  between  its  extreme  dates,  the  difference  between  the  Interest 
and  discount  of  the  Dr.  items  from  their  respective  dates  to  the  due  date  (by  equation)  must 
be  offset  or  balanced  by  the  difference  between  the  interest  and  discount  of  the  Cr.  items,  from 
their  respective  dates  to  the  due  date,  within  one-half  of  the  interest  or  discount  on  the  balance 
for  one  day. 


EQUATION   OF   ACCOUNTS. 


253 


805.     Example. — What  is  the  balance  of  the  following  account,  and  when  is 
it  due  by  equation  ? 


Di 


1886. 

Feb. 

1 

ii 

10 

To  Mdse., 


(Student's  Ledger.) 
Charles  S,  Williams. 


Cr. 


1886 

600 

Feb. 

19 

1800 

" 

28 

Mar. 

6 

Bv  Cash, 


300 
300 
300 


Operation. 
Dr. 
.luue  19,  1886, /om/  date. 

1886.  Feb.     1,  %  600.     138  days  to  focal  date  =  $13.80,  interest. 
•'       Feb.  10,    1800.     129    '••     ''     "        "    =    38.70, 


Total  Dr., 

$2400. 

886.   Feb.  19, 

$300. 

"      Feb.  28, 

300. 

Mar.    6, 

300. 

Total  Cr. 

$900. 

$52.50,  total  Dr.  interest. 


Cr. 


120  days  to  focal  date  =    $6.00,  interest. 
Ill     '''     "     "        "     =      5.55,        " 
105     "     "     ''        "     =      5.25.        " 

$16.80,  total  Cr.  interest. 

$52.50,  Dr.  interest. 
Dr.  balance  $1500.  16.80,  Cr.  interest. 

Interest  of  $1500  for  1  day  =  |  .25.  $35.70,  excess  Dr.  interest. 

$35.70  -^  $.25  =  142^  =  143  days  equated  time. 
143  davs  back  from  June  19  =  Jan.  27,  1886. 


Remark. — Since  debit  and  credit  accounts  are  accounts  wherein  both  debtor  and  creditor 
are  represented  by  certain  purchases  (debts)  and  payments,  and  since  the  items  constituting  the 
Dr.  on  the  Lodger  of  one  of  the  parties  would  constitute  the  Cr.  on  the  Ledger  of  the  other 
partj',  and  vice  ncrm,  it  follows  that  an  account  equated  from  both  these  views  must  show 
like  conclusions;  i.  e.,  the  above  account  reversed,  so  that  its  Cr.  shall  appear  a  Dr.,  and  its 
Dr.  appear  a  Cr.,  and  equated  from  any  date  as  a/<?co^date,  must  show  the  same  conclusion 
as  before. 


FiXA.MPJj;. — Same  as  before,  reversed,  and  with  May  1  assumed  as  a  focal  date. 


Dr. 


(Charles  S.  Williams'  Ledger.) 
"Student." 


Cr. 


1886. 

Feb. 

19 

(( 

28 

Mar. 

6 

1886. 

To  Cash, 

300 

Feb. 

(<      << 

300 

a 

"      " 

300 

1 

10 


By  Mdse., 


600 
1800 


264  EQUATION"  OF   ACCOUNTS, 

Operation. 

Dr. 

May  1,  1886,  focal  date. 

188*6.  Feb.  19,  *300.       71  days  to  focal  date  =  *3.55,  interest. 
".     Feb.  28,    300.       62     '''     "     "       ''     =    3.10, 
"       Mar.    6,    300.       56     "     "     "       "     =    2.80. 

Total  Dr..  $900.  ?!9.4o,  total  Dr.  interest. 

Cr. 

1886.  Feb.    1,  *  600.     89  days  to  focal  date  =  %  8.90,  interest. 
"       Feb.  10,     1800.     80     "     "     "       "     =    24.00, 
Total  Cr..  §2400.  $32.90,  total  Cr.  interest. 

9.45,  total  Dr.  interest. 

Cr.  balance,  $1500.  $23.45 

Interest  of  $1500  for  1  day  =  $.25. 

$23.45  -^  $.25  =  93 i  =  94  days  equated  time. 

94  days  back  from  May  1,  1886  =  Jan.  27,  1886,  as  before  found. 

Example  (same  as  first  illustrated). — Proof,  assuming  Jan.  27,  1886,  as  a 

focal  date. 

Operation. 

Dr. 
1886.   Feb.     1,  $  600.       5  days  after  focal  date  =  $  .50,  discount. 
"       Feb.  10,    1800.     26     ^"'       •••         ••        ••'     =    4.20, 

4.70,  total  Dr.  discount. 
Cr. 

1886.   Feb.  19,    $300.     23  days  after  focal  date  =  $1.15,  discount. 
"       Feb.  28.      300.     32     •'•'       "        "       "     =    1.60, 
"       Mar.    0,      300.     38     "       "         "       "     =    1.90, 

$4.65,  total  Cr.   discount. 

Cr.  balance,  $1500.  $4.70,  total  Dr.  discount. 

Discount  on  $1500  f(.r  1  day,  $.25.  4.65,     "      Cr. 

$.05,  difference. 

Explanation. — The  difference  between  the  Dr.  discount  and  the  Cr.  discount  is  5  cents, 
or  ,*5  =  i  of  the  discount  on  the  $1500  balance  for  1  day,  or  less  than  one-half  of  1  day's 
discount,  thus  proving  the  balance  to  have  been  due  since  Jan.  27, 1886,  as  determined  by  both 
the  former  operations,  and  rendering  an  explanation  which  could  be  made  in  the  usual  form 
quite  unnecessary. 

Rule. — Find  the  face  hahnice  of  the  account,  and  also  the  excess  of 
interest  from  the  latest  date  as  a  focal  date.  If  the  halance  of  account 
and  excess  of  interest  he  on  the  sai)ie  side,  date  hack;  if  on  opposite 
sides,  date  forward. 


4 


EXAMPLES   IN    EQUATION    OF   ACCOINTS.  255 

EXAMPLKS   FOR   PRACTICE. 

8(HJ.     i.     When  is  tlie  balance  of  tlie  following  account  due  by  equation  ? 
[)r.  Frank   H.  Barxard.  Cr. 


1887. 

Jan. 
Feb. 


15 

28 


To  Mdse., 


i  i      1887. 

600 

1   Feb. 

1 

300 

1    Mar. 

31 

By  Cash, 


300 
300 


2.     What  is  the  balance  of  the  following  account,  and  wjien  due  by  CM[uaiiun  ? 
Dr.  Benj.  F.   Hawkins.  Cr. 


1887. 

Jan. 

14 

<( 

28 

Feb. 

3 

(( 

15 

To  Mdse., 


a  (( 

it  sf 


1887. 

600 

i 

Jan. 

20 

300 

1 

Feb. 

10 

500     1 

600 

By  Cash, 


1000 
700 


S.     If  money  be  worth  7^  per  annum,  what  was  the  cash  balance  due  on  the 
following  account  Julv  1,  1887  ? 


Dr. 


Victor  E.  Brown  &  Co. 


Cr. 


1887. 

Jan. 

31 

Mar. 

30 

To  Mdse., 


i(  a 


1887. 

1 

450 

Jan. 

2 

450 

Feb. 

13 

\ 

Mar. 

29 

By  Mdse., 
"    Cash, 
''  Mdse., 


(iOOJ 
3001 
300l 


Jf.     What  was  the  cash  balance  duo  on  the  following  account  Jan.  1,  1889,  if 
money  l)e  worth  8<^  per  annum  ? 

Dr.  Henry  J.  Sanford  &  Bro.  Cr. 


1888. 

Aug. 
Sept. 
Oct. 
Dec. 


4i  To  Mdse.,  1  mo., 
1      "       •'        2  mo., 
31 


li  iC 


2 

4  mo.. 


1888. 

200 

Oct. 

1 

400 

Nov. 

1 

600 

Dec. 

1 

300 

1889. 

Jan. 

1 

Feb. 

1 

Mar. 

1 

By  Cash, 


a  li 


150 
150 
150 

150 
150 
150 


r>.     Fijid  \\\v  balance  of  the  following  account,  and  when  due  by  equation. 
Dr.  Louis  K.  Gould.  Cr. 


1888. 

Sept. 

21 

To  Mdse. 

Oct. 

5 

a          ic 

(< 

30 

a         a 

Dec. 

18 

<i           a 

:889. 

Jan. 

31 

i.           " 

Feb. 

28 

<<          a 

60  da., 
30  da., 
60  da., 

1  mo.. 


1888. 

100 

Nov. 

1 

150 

(( 

28 

116 

50 

Dec. 

31 

251 

45 

1889. 

Jan. 

15 

80 

75 

Mar. 

1 

100 

10 

By  Cash, 

••    Mdse.,  1  mo., 

••       *'        2  mo., 

■•    ("ash. 


70 
110 
120 

175 
200 


50 


/J56 


EXAMPLES    IX    EQUATION    OF    ACCOUNTS. 


6".     What  is  the  balance  of  the  following  account,  and  when  due  by  equation  ? 
Dr.  Reed  &  Co.  Cr. 


1888. 

1888. 

June 

14 

To  Mdse., 

300 

Julv 

1 

By  Cash, 

100 

<< 

29 

'*    Cash, 

150 

Aug. 

1 

i(         e< 

100 

Aug. 

4 

"   Mdse., 

200 

Sept. 

1 

CC              <( 

100 

Oct. 

31 

*'    Cash, 

100 

Oct. 

1889. 

Jan. 

1 

1 

"■   Mdse., 

100 
450 

Remark. — Interest  may  be  computed  on  one  of  the  four  similar  Cr.  items  for  the  aggregate 
of  their  davs. 


7.     When  is  the  balance  of  the  following  account  due  by  equation  ? 
Dr.  King  &  Sherwood. 


Cr. 


1888. 

Nov. 

3 

Dec. 

31 

1889. 

Jan. 

11 

Mar. 

4 

1888. 

750 

Dec. 

20 

1000 

1?89. 

Jan. 

1 

600 

Feb. 

1 

150 

May 

3 

To  Mdse.,  750  Dec.    20    Bv  Cash, 

1000  1589. 

"  Mdse., 

"   Cash, 


8.     When  is  the  balance  of  the  following  account  due  by  equation  ? 
Dr.  Samuel  Peck  &  Sox. 


500 

500 

1500 

500 


Cr. 


1887. 

1887. 

Mar. 

3 

To  Mdse., 

60 

Apr. 

1 

By  Cash, 

150 

Apr. 

24 

<<       (( 

100 

June 

1 

<<          a 

150 

May 

1 

((       i( 

150 

Aug. 

1 

a          a 

150 

Aug. 

30 
17 

90 
:  200 

Oct. 

1 

a          11 

90 

9.     Find,  1st,    the    balance   of    the    following   account;    2d,    when   due   by 
equation. 
Dr.  Walter  L.  Parker.  Cr. 


1888. 

1888. 

May 

11 

To  Mdse. 

2  mo.. 

'  108 

40 

June 

1 

By  Cash, 

124 

27 

Julv 

1 

n         n 

30dii., 

1  225 

!  Oct. 

31 

"   4  mo.  note  (no 

Aug. 

31 

li          a 

280 

80 

1 

interest). 

167 

91 

Oct. 

1 

i(          a 

1  137 

50 

1  Dec. 

1 

"   Cash, 

306 

05 

10.     Find,  1st,  when  the  following  account  is  due  by  equation  ;   2d,  the  cash 
balance  due  Jan.  1,  1888,  if  money  be  worth  5^  per  annum.     Prove  the  result. 

Dr.  John  Montgomery  &  Co.  Cr. 


1887. 

Dec. 


1888. 

Jan. 


15 

28 

14 


To  Mdse., 

"       ''        2  mo., 

"       "        30  da.. 


200 
300 

300 


1888 

Jan. 
Mar. 


By  Cash, 
"    60-da, 


note  (no 
interest). 


300 
150 


EXAMPLES   IN"    EQUATION   OF   ACCOUNTS. 


257 


Remark.— In  case  a  negotiable  paper  is  given,  its  maturity  is  determined  in  the  usual 
way,  by  adding  to  its  express  time  three  days  of  grace.  If  tlie  paper  bear  interest,  its  value 
is  equivalent  to  ilsface  as  cash  at  its  date;  while  if  the  paper  be  non-interest  bearing,  its  value 
is  equivalent  to  cash  at  its  full  maturity. 

11.    Find,  1st,  the  balance  of  the  following  account;  2d,  when  due  by  equation; 
3d,  cash  balance  due  Jan,  1,  1888,  if  money  be  worth  6;^  per  annum.     Prove  the 
result. 
Jjr.  E.  E.  EoGERS  &  Bro.  Cr. 


1887. 

May 

14 

June 

3 

July 

31 

To  Mdse.,  1  mo., 
"  ''  60  da., 
"       "        2  mo., 


1887. 

300 

May 

31 

200 

400 

July 

1888. 

15 

Jan. 

1 

By  2-mo.  note  (no 

interest), 
'*'   30-da.  note,  on 

interest, 

''    Cash, 


240 
150 
100 


1^.    Find,  1st,  the  balance  of  the  following  account;  2d,  when  due  by  equation; 
3d,  the  cash  balance  due  Jan.  1,  1888,  if  money  be  worth  10^  j^er  annum.     Prove 
the  result. 
Dr.  King,  Son  &  Co,  Cr. 


1887. 

Oct. 

] 

Nov. 

3 

Dec. 

14 

1888. 

Jan. 

15 

1887. 

150 

Nov. 

1 

150 

Dec. 

1 

300 

1888. 

300 

Feb. 

15 

To  Mdse.,  1  mo.,  150       |    Nov.      1     By  Cash, 

2  mo.,  150  Dec.      1      "    3-mo,  accpt.  (no 

60  da,,         300  interest), 

'  •    Cash, 

lo.     When  is  the  balance  of  the  following  account  due  by  equation  ? 
Dr.  Spaulding  &  Co, 


200 
200 

200 


Cr. 


Oct. 
(( 

1889. 

Jan. 
Feb. 


14 


To  Mdse.,  30  da,, 
"       "        4  mo.. 


60  da. 


1888. 

278 

50 

Nov. 

20 

147 

50 

Dec. 

31 

100 

25 

1889. 

311 

50 

i 

Mar. 

1 

By  Cash, 

"   2-mo.  accpt,  (no 
interest), 

'*    60-da,  note,  on 
interest. 


210 
175 

220 


50 


14.  Find,  1st,  the  balance  of  the  following  account;  2d,  when  due  by  equation; 
3d,  the  cash  balance  due  Mar,  1,  1880,  if  money  be  Avorth  o^  i)er  annum.  Prove 
the  result. 


Dr. 


Abraham  Bradley, 


Cr. 


1888, 

Aug, 

31 

Sept, 

5 

Oct. 

31 

Dec. 

19 

1889. 

Jan. 

1 

By  Mdse.,  1  mo., 
"  "  60  da., 
"  "  4  mo., 
"       "        30  da.. 


<e 


1  mo.. 


1888, 

150 

Oct. 

2 

200 

600 

(( 

30 

150 

Dec. 

1 

100 

1889. 

Jan. 

25 

17 


By  30-da.  note  (no 

interest), 
''   Cash, 
''   60-da.  note,  on 

interest, 

'■    1-mo.  accj)t.  (no 
interest). 


100 
200 

300 


500 


258 


EXAMPLES   IN   EQUATION   OF  ACCOUNTS. 


15.  Find,  1st,  the  balance  of  the  following  account ;  2d,  wlien  due  by  equa- 
tion ;  3d,  the  cash  balance  due  Apr.  1,  1889,  if  money  be  worth  7^  per  annum. 
Prove  the  result. 


Dr 


Lee  cS:  Powers. 


Cr.. 


1888. 

Sept. 

9 

Oct. 

1 

Dec. 

13 

1889. 

Jan. 

31 

To  Mdse., 


2  mo., 
1  mo., 

1  mo., 


1889. 

600 

Jan. 

1 

300 

1 

Mar. 

IG 

150 

Aiu'. 

30 

450 

May 

1 

By  Cash, 

"   2-mo.  note,  on 

interest, 
''   3-mo.  note  (no 

interest), 
"    Cash, 


500 

100 

300 
200 


16.     When  are  the  net  proceeds  of  tlie  following  account  sales  due  by  equation  ? 

Kansas  City,  Mo.,  Oct.  3,  1888. 
Account  Sales  of  Flour, 

Sold  for  account  of  Henry  H.  Grinnell  &  Co., 

Burlington,  Iowa. 
By  C.  H.  Brayton. 


1888. 

Sept. 

23 

Oct. 

1 

a 

18 

Nov. 

3 

n 

25 

Sept. 

24 

a 

•ZC) 

Oct. 

28 

Nov. 

15 

(( 

25 

95  barrels  to  Hudson  &  Son, 


200 

65 

110 

130 


"  Chas.  II.  Knapp, 


"  Wm.  Clark  &  Bro., 
"  Clinton  McPherson, 
Charges. 

Freight, 

Cartage, — 

Cash  advanced  on  consignment,  . 

Cooperage, - 

Commission,  4^, 


@  $5.60,  cash, 
@  $5.75,  1  mo., 
@  $5.80,  60  da., 
@  $5.80,  30  da., 
@  $5.75,  casii, 


62 

30 

2000 

5 

137 


Remarks. — 1.  In  rendering  Accounts  Sales,  the  expenses  ( freight,  storage,  commission,  etc.) 
charged  constitute  the  Debits  of  the  account,  while  the  gross  sales^onstitute  thfc  Credits.  Equate 
such  accounts  in  tlie  usual  manner. 

2.  After  extension  of  time  to  determine  actual  due  (or  just  Cr.)  dates  of  the  items  on  both 
sides  of  the  account,  should  it  then  be  found  that  certain  items  of  the  Dr.  have  dates  corres- 
ponding to  those  of  certain  items  of  the  Cr.,  such  items,  if  of  equal  amount,  may  be  cancelled 
the  one  against  the  other;  if  of  unequal  amounts,  they  may  be  offset  for  like  amounts,  and 
only  their  difference  enter  into  the  work  of  the  equation. 


RATIO.  2o!» 


RATIO. 

807.  Ratio  is  the  relation  of  one  number  to  another  of  the  same  denomi- 
nation.    It  is  of  two  kinds,  Arithmetical  and  Geometrical. 

808.  Arithmetical  Ratio  is  the  difference  of  the  two  numbers;  as,  the 
arithmetical  ratio  of  7  and  3,  or  7  —  3  =  4. 

Remark. — Arithmetical  ratio  indicates  subtraction,  and  is  or  shows  a  difference. 

809.  Geometrical  Ratio  is  the  quotient  of  one  number  divided  by  another: 
as,  the  ratio  of  G  to  2,  or  0  -^  2  =  3. 

810.  The  Sign  of  Ratio  is  the  colon  (:),  and  is  considered  to  be  the 
division  sign  Avith  the  horizontal  bar  omitted,  and  is  read  is  to.  Thus,  G  :  2  is 
read,  6  is  to  2. 

811.  The  Terms  of  a  ratio  are  the  two  numbers  compared,  and  taken 
together  they  are  called  a  couplet. 

81*2.  The  left  hand  term  of  an  arithmetical  ratio  is  called  the  antecedent,  and 
stands  in  the  relation  of  a  minuend;  the  right  hand  term  is  called  the  consequent, 
and  stands  in  the  relation  of  a  subtrahend. 

813.  In  geometrical  ratios,  the  antecedent  corresponds  to  the  dividend,  and 
the  consequent  to  the  divisor;  and  to  such  ratios  the  General  Principles  of 
Division  appl}",  as  follows: 

1st.  Any  change  in  the  antecedent  produces  a  like  change  in  the  ratio. 
2d.  Any  change  in  the  consequent  produces  an  opposite  change  in  the  ratio. 
3d.  A  similar  change  effected  in  both  terms  will  not  change  the  ratio. 

814.  Reverse,  indirect,  or  reciprocal  ratios  are  formed  by  reversing  the 
position  or  order  of  the  terms. 

815.  Simple  Ratio  is  the  ratio  of  two  numbers;  as  20  :  5. 

816.  Compound  Ratio  is  the  ratio  of  the  products  of  the  corresponding 
terms  of  two  or  more  ratios;  as,  20  :  5  and  15  :  3  may  be  compounded  and  read 
20  X  15  :  5  X  3,  which,  Avhen  the  multiplication  is  performed,  becomes  a  simple 
ratio. 

{Ratio  =  Antecedent  —  Consequent. 
Consequent  =  Antecedent  —  Ratio. 
_         ,        ,  Antecedent  =  Consequent  +  Ratio. 

Jformulas.  \ 

{Ratio  =  Antecedent  -^  Consequent. 
Consequent  =  Antecedent  -^  Ratio. 
Antecedent  =  Consequent  x  Ratio. 


260  SIMPLE    PROPOBTIOX. 


PROPORTION. 

817.     Proportion  is  an  equality  of  ratios,,  and  is  indicated  in  two  ways: 
1st.   By  placing  the  sign  of  equality  between  the  ratios;  thus,  8  :  2  =  12  :  3;  or, 
2d.  By  placing  a  double  colon  ( :  : )  between  the  ratios;  thus,  8  :  2  : :  12  : 3, 
which  reads,  8  is  to  2  as  12  is  to  3. 

Remarks. — 1.  The.^rs^  sind  fourth,  or  outside  terms,  of  a  proportion  are  called  the  extremes; 
the  second  and  third,  or  inside  terms,  are  called  the  means. 

2.  Observe  that,  in  the  arithmetical  proportion,  7 :  3  : :  12  :  8,  the  sum  of  the  extremes  equals 
the  sum  of  the  means.  If,  then,  either  extreme  be  wanting,  it  may  be  found  by  subtracting 
the  gircn  extreme  from  the  sum  of  the  means.  If  either  mean  be  wanting,  it  may  be  found 
by  subtracting  the  given  mean  from  the  sum  of  the  extremes.  If  the  ex'tremes  be  equal  and 
both  wanting,  each  must  equal  one-half  of  the  sum  of  the  means,  and  if  the  means  be  equal, 
and  both  wanting,  each  must  equal  one  half  of  the  sum  of  the  given  extremes.  This  is  shown 
and  its  use  made  valuable  in  proportions  of  three  terms;  a.s,  9  :  6  : 3,  in  which  6  is  a  mean 
proportional  term,  the  extended  form  being  9  :  6  : :  6  :  3. 

3.  Observe  that,  in  the  geometrical  proportion,  12  :  4 :  :  15  :  5,  the  product  of  the  extremes 
equals  the  product  of  the  means.  If,  then,  either  extreme  be  wanting,  and  both  means  given, 
the  wanting  extreme  can  be  foimd  by  dividing  the  product  of  the  means  by  the  given  extreme; 
and  if  one  mean  be  wanting,  and  both  extremes  given,  the  wanting  mean  can  be  found  by 
dividing  the  product  of  the  extremes  by  the  given  mean.  And  if  the  extremes  be  equal  and 
both  wanting,  each  must  equal  the  square  root  of  the  product  of  the  means;  and  if  the  means 
be  equal,  and  both  wanting,  each  one  must  equal  the  square  root  of  the  product  of  the  extremes. 
This  is  again  shown  and  its  use  made  valuable  in  proportions  of  three  terms;  as,  27  :  9  :  3,  in 
which  9  is  a  mean  proportional  term,  the  extended  form  being  27  :  9  : :  9  :  3. 


SIMPLE    PROPORTION. 

818.  A  Simple  Proportion  is  an  equality  of  two  simple  ratios;  thus, 
27  : 3  :  :  45  : 5,  consisting  of  four  terms,  the  relations  of  which,  as  above  explained, 
are  such  that,  if  any  three  of  them  are  given,  the  fourth  may  ^eadil}'  be  found; 
for  this  reason,  solutions  by  proportion  were  said,  by  the  old  writers,  to  come 
under  "the  Rule  of  Three.'' 

819.  Take  the  proportion  27  :  3  :  :  45  :  5,  and  suppose  the  last  extreme 
unknown,  and  indicate  its  value  by  a;.  The  proportion  would  read,  27  :  3  : :  45  :  a;, 
in  which  the  value  of  x  is  found  by  dividing  3  X  -45  (the  product  of  the  means) 
by  27  (the  given  extreme);  3  X  45  =  135;  135  -=-  27  =  5;  hence,  rr  =  5. 

Rules.— i.  Divide  the  product  of  the  given  means  by  the  given  extreme; 
the  quotient  will  be  the  other  extreme.    Or, 

2.    Divide  the  product  of  the  given  extremes  by  the  given  mean ;  the 

qiwtiejit  mill  he  the  other  mean. 

Remark. — Since  the  unknown  term  and  its  given  multiplier  (mean  or  extreme)  constitute 
the  factors  of  the  divisor,  and  the  remaining  two  terras  the  factors  of  the  dividend,  the  rules 
for  CvNCELLATiON  apply,  and  their  use  will  simplify  the  work. 


COMPOUND    PROPORTION.  261 

EXAMPLES   FOK   PRACTICE. 

820.     Find  the  unknown  term  in  each  of  the  following  proportions: 


/.     39  :  3  :  :  52  :  a;. 

2.  105  :15  :  :.'c:  4. 

3.  42  :  .T  :  :  54  :  9. 


Jf.     a; :  9  :  :  45  :  5. 

5.  96  yd.  '.x::  $134.50  :  *403.50. 

6.  .T  :  177.50  :  :8bu.  2 pk.  :  153 bu. 

7.  If  a  post  7i  ft.  !iigh  casts  a  shadow  1^  ft.,  what  is  the  hight  of  a  tower 
that  casts  a  shadow  150  ft.  at  the  same  time? 

8.  If  15  bushels  of  wheat  can  be  bought  for  $13.50,  how  many  bushels  can 
be  bought  for  $430.20? 

9.  An  insolvent  debtor  owes  $14400,  and  has  an  estate  valued  at  $10800. 
How  much  will  A  receive,  on  a  claim  of  $3750? 

10.     A  friend  loaned  me  $750,  for  3  yr.  4  mo.  15  da.     For  what  period  of 
time  should  I  loan  hini  $900  to  fully  repay  his  favor? 


COMPOUND    PROPORTION. 

821.  A  Compound  Proportion  is  a  proportion,  any  of  the  terms  of  which 
have  been  compounded — /.  e.,  in  which  such  terms  are  made  up  of  factors;  as,  the 
simple  proportions  6  :  2  :  :  15  :  5  and  21  :  3  :  :  28  :  4,  become  compound  when 
expressed  6  X  21  :  2  X  3  ::  15  X  28  :  5  X  4.     This  is  more  conveniently  expressed 

as  follows: 

6. 2.. 15.  5 
21  -3 ••28-4 

Remark. — lu  compound  proportions,  wherein  the  number  of  factors  in  the  couplets  are 
more  than  two,  it  is  well  to  substitute  lines  for  colons,  as  follows: 


6  I  2 
21     3 


15 

28 


822.  Every  question  of  proportion  involves  the  principle  of  cause  and  effect. 
That  is,  work  done  for  pay,  cash  given  for  goods,  wood  cut  by  labor  performed, 
investments  made  resulting  in  gains  or  losses,  etc.;  and  to  keep  theorizing  as 
simple  as  possible  in  a  subject  rarely  used,  it  seems  best  to  adhere  to  some  one 
of  the  many  logical  statements  of  the  principles  of  proportion,  as  follows: 
•      1st  Cause  :  1st  Effect  :  :  2d  Cause  :  2d  Effect. 

This  will  apply,  whichever  term  may  be  unknown,  and  will  apply  as  well  to 
groups  of  causes  or  groups  of  effects,  as  maybe  shown  in  compouiul  proportions. 
Take  for  illustration  the  following: 

Example. — If  10  men,  working  12  days,  of  8  hours  each  day,  can  cut  200 
cords  of  wood,  how  many  cords  should  be  cut  by  13  men  in  15  days,  if  they  work 
6  hours  per  day? 

Explanation. — Observe  that  the  cutting  of  200  cords  of  wood  is  an  effect  produced,  the 
cause  of  which  was  10  men,  working  for  12  days  of  8  hours  per  day;  and  that  the  working  of 
12  men  for  15  days  of  6  hours  \>er  day  was  a  cause,  the  effect  of  which  is  unknown;  but  from 
the  application  of  the  logical  statement  of  the  principles  of  proportion  (1st  Cause  :  1st  Effect : : 
2d  Cause  :  2d  Effect),  we  have  the  statement  of  the  example  given  in  form  as  follows: 


20- 


COMPOUND    PROPORTION. 


IstCaust'.    1st  Effect.      2d  Cause.   2d  Effect.  And  since,  as  before  shown,  the  extreme  (or 

10  1        ^'"^  outside)  terms  constitute  the  factors  of  the  divisor, 

12  200  1^  ^  and  the  mean  (or  inside)  terms  constitute  the  fac- 

8  j  6  tors  of  the  dividend,  any  factor  of  the  divisor' may 

be  cancelled  against  any  factor  of  the  dividend, 
or  vice  versa.     Reproducing  the  above  statement,  and  effecting  possible  cancellations,  we  have: 

:  225  cords. 
5  X  15  X  3  =  225. 

Remark. — All  problems  in  proportion,  simple  or  compound,  by  some  called  the  "single 
rule  of  three"  or  the  "  double  rule  of  three,"  can  be  solved  as  above. 


10 

n 

n 

m 

15 

H 

^ 

0 

3 

kJcamplks  for  practick. 

823.  1.  If  5  men,  working  G  days  of  12  hours  per  day,  can  cut  2-4  acres  of 
com,  how  many  acres  of  corn  should  8  men  cut  in  5  days,  if  they  work  10 
hours  per  day? 

2.  If  6  men,  working  for  12  days,  dig  a  ditch  80  rods  long,  how  many  rods 
of  such  ditch  should  15  men  dig  in  21  days? 

Remark. — When  any  term  or  terms  is  fractional,  either  common  or  decimal  in  form,  treat 
them  in  the  usual  manner,  or  reduce  such  fractions  to  a  common  denominator  and  compare 
their  numerators. 

S.  If  15  men  earn  $607.50  in  18  days,  how  much  should  21  men  earn  in  12 
days? 

4.  If  $1600,  invested  in  a  business  for  3  years,  gain  $900,  liow  much  should 
$2150  gain  in  the  same  time. 

5.  If  8145.35  interest  accrue  on  $510,  at  6f/,  in  4  yr.  9  mo.,  how  much  interest 
will  accrue  at  the  same  rate  and  time  on  $1350? 

6'.  If  40  yards  of  carpet,  f  of  a  yard  in  widtli,  Avill  cover  a  room  18  feet  long 
and  15  feet  wide,  how  many  yards  of  carpet,  \  of  a  yard  in  widtli,  will  cover  a 
room  35  feet  long  and  28  feet  in  width? 

7.  If  $684,  at  interest  for  3  yr.  3  mo.  18  da.,  at  5fc,  accrue  $112.86  interest, 
at  what  rate  per  cent,  must  $1800  be  put  at  interest  for  the  same  time  to  accrue 
$445.50  interest? 

8.  If  $760,  put  at  interest  at  10«s5,  accrue  $9.50  interest  in  45  days,  in  how 
many  days  will  $1140  accrue  $17.67  interest  at  <o<  ? 

Remark. — The  subjects  of  Ratio  and  Proportion  have  been  briefly  discussed  as  above 
for  fhc  sole  purpose  of  the  introduction  and  use  of  the  analysis  of  the  principles  involved  in 
them,  in  the  division  of  the  gains  or  losses  in  partnerships. 


PARTNERSHIP.  263 


PARTNERSHIP. 

824.  Partnership  is  the  association  resulting  from  an  agreement  between 
two  or  more  persons  to  place  their  money,  effects,  labor,  and  skill,  or  some  or  all 
of  them,  in  some  enterjirise  or  business,  and  divide  the  profits  and  bear  the  losses 
in  certain  proportions. 

825.  Partnerships  may  be  formed  by  written  agreement,  sealed  or  unsealed, 
by  oral  agreement,  or  by  implication. 

Remark. — Important  partnerships  should  be  formed  by  written  agreements,  in  which  all  of 
the  conditions  of  the  partnership  should  be  fiilly  slated. 

826.  The  business  association  is  generally  called  a  Firm,  but  is  sometimes 
'designated  as  a  House. 

827.  The  Capital  consists  of  the  money  or  other  property  invested. 

828.  The  Resources  or  Assets  of  a  firm  consist-of  the  property  it  owns  and 
the  debts  due  the  firm. 

829.  The  Liabilities  of  a  firm  are  its  debts. 

830.  The  Net  Capital  is  the  amount  which  the  resources  exceed  the  liabilities. 

831.  The  Net  Insolvency  is  the  amount  which  the  liabilities  exceed  the 
resources. 

832.  The  Net  Investment  of  a  partner  is  the  amount  of  the  firm's  capital 
whicli  he  has  invested,  less  the  amount  which  lie  may  have  withdrawn  from  the 
business. 

833.  The  Net  Grain  is  the  excess  of  the  total  gains  over  the  total  losses,  for 
a  given  period. 

834.  The  Net  Loss  is  the  excess  of  the  total  losses  over  the  total  gains,  for 
a  given  period. 

835.  Partners  are  of  four  classes: 

1.  Heal  or  ostensible. 

2.  Dormant,  silent,  or  concealed. 

3.  Limited. 

4.  Nominal. 

836.  A  Real  or  Ostensible  Partner  is  one  wlio  appears  to  the  world  to 
be  and  who  actually  is  a  partner. 

837.  A  Dormant  or  Silent  Partner  is  one  whose  name  does  not  appear 

in  the  firm  name,  whose  relation  is  puri)osely  concealed,  but  who  yet  profits  by 
an  investment. 

Remark. — The  rule  concerning  silent  partners  is,  that,  being  sharers  in  the  firm's  profits, 
they  are  liable  the  same  as  real  partners  to  all  creditors  of  the  firm  who,  either  before  or  after 
irusting  the  firm,  learn  of  their  connection  therewith. 


264  PARTNERSHIP. 

838.  A  Limited  Partner  is  one  who,  according  to  the  requirements  of 
statute  law,  publishes  his  connection  with  the  firm,  names  the  limit  of  his  respon- 
sibility thereby  assumed,  and  in  that  manner  escapes  general  responsibility. 

839.  A  Nominal  Partner  is  one  whose  name  appears  to  the  public,  but 
who  has  no  investment,  and  receives  no  share  of  the  gains. 

Remark. — The  rule  of  law  concerning  nominal  partners  is,  that  false  .appearances  have 
been  held  out  by  them,  and  that  all  persons  trusting  the  tirni,  on  account  of  the  association  of 
their  names  with  it,  are  entitled  to  hold  them  the  same  as  if  they  were  real  partners. 

840.  To  Divide  the  Gain  or  Loss,  when  each  Partner's  Investment  has  been 
Employed  for  the  Same  Period  of  Time. 

Remark.  —In  determinning  the  division  of  the  gains  or  losses  in  partnership,  the  principles 
of  I*roportion  will  be  found  applicable,  as  in  the  following: 

Example. — A  and  B  together  bought  a  house  for  $8750,  of  which  A  paid 
$5000,  and  B  paid  $3750.  If  the  house  rents  for  $560  a  year,  how  many  dollars 
of  the  rent  shoiild  each  receive? 

•Remark. — By  reference  to  conditions  heretofore  given,  it  will  be  observed  that  money 
invested  is  a  cause,  and  profit  therefrom  is  an  effect. 

From  the  above  example,  we  have  the  following 

Statement. 

Investment  of §8750,  first  cause. 

Gain,  in  rent,  of 560,  first  effect. 

A's  investment  of 5000,  second  cause. 

The  unknown  term second  effect. 

From  which  relations  we  have,  by  application  of  the  principles  and  use  of  the  explained 
forms  of  Proportion,  the  following 

Operation. 

IstC.      IstE.         2d  a  2dE. 

$8750  :  560  :  :  5000  :  (A's  part  of  the  rent). 
Reproducing  and  canceling,  we  have: 

$rin  :  060  :  :  0000  :  (A's  part). 
:?  8         40 

8  X  40  =  *320,  A's  part  of  the  rent. 
B's  part  is  the  difference  between  the  whole  rent,  $560,  and  the  part  to  which  A  is  shown  Xx> 
be  entitled;  or  it  may  be  obtained  by  application  of  the  same  form  as  that  used  to  determine  the 
gain  of  A,  viz. : 

$n^  :  060  :  :  3^00  :  (B's  part). 
;?         8       30 
8  X  30  =  $240,  B's  part  of  the  rent. 

Rule. — The  ichdle  capital  is  to  the  whole  gain,  as  each  partner  s  sTiare 
of  the  capital  is  to  his  share  of  the  gain. 

Remarks. — 1.  Should  the  result  of  the  investment  be  a  los.s,  the  share  to  be  sustained  by 
each  can  be  determined  in  the  same  manner  as  above. 

2.  If  investments  are  made  for  different  periods  of  time,  compute  the  investment  of  each 
partner  for  one  period  of  that  time,  day,  month,  or  year,  then  make  the  proportion  as  above. 


EXAMPLES   IN    PARTNERSHIP.  265 

EXAMPLES   FOK   PRACTICE. 

841.  i.  Two  men  boiight  a  mine  for  120000,  of  which  sum  A  paid  $12500, 
and  B  paid  the  remainder;  they  afterwards  sold  the  mine  for  142000.  How  much 
of  the  selling  price  was  each  partner  entitled  to  receive? 

2.  The  condition  of  the  business  of  Hadley  &  Hunt  is  as  follows:  Mdse.  OD 
kand,  $28240;  notes  and  accounts  due  the  firm,  121416.54;  cash  on  hand, 
$1619.62;  total  liabilities  of  the  firm,  $23186.75.  Hadley's  investment  was 
$9000,  and  Hunt's  $12500.  What  has  been  the  gain  or  loss,  and  what  is  the 
share  of  each? 

3.  A,  B,  C,  and  D,  engaged  in  a  business,  in  which  D  invested  88400,  which 
was  also  the  amount  of  the  net  gain;  if  A's  share  of  the  gain  was  $1800,  B's 
$3000,  and  C's  $2400,  what  must  have  been  the  whole  capital  and  D's  gain? 

4.  A,  B,  and  C  are  partners,  A's  investment  being  $9600,  B's  $8100,  and  C's 
$7500.  At  the  end  of  the  year  they  have  resources  amounting  to  $27850,  and 
liabilities  amounting  to  $3150.  What  is  the  present  worth  of  each  partner  at 
closing? 

5.  Four  partners.  A,  B,  C,  and  D,  invested  equal  amounts,  and  agreed  ta 
equally  ajiportion  the  gains  or  losses.  At  the  time  of  dissolution,  the  firm  had 
resources  to  the  amount  of  $33800,  and  liabilities  to  the  amount  of  $51975.  If 
the  net  loss  was  $27460,  what  Avas  the  net  insolvency  of  each  partner  at  the  time 
of  dissolution?  What  was  each  partner's  investment  ? 

6.  A  and  B  were  partners  1  year,  each  investing  $3500,  and  agreeing  to 
equally  share  the  gains  or  sustain  the  losses.  At  the  close  of  the  year  their 
resources  were:  Cash,  $2650;  Mdse.,  $3040;  accounts  due  them,  $3150.  During 
the  year,  A  drew  out  $4500,  and  B  $5750.  How  much  has  been  gained  or  lost? 
What  is  the  solvency  or  insolvency  of  the  firm?  What  is  the  present  worth 
of  each? 

7.  Harrison  and  Morton  bought  a  section  of  Nebraska  jirairie  for  $8000, 
Harrison  paying  $5000,  and  Morton  paying  the  remainder.  Cleveland  offered 
them  $8000  for  one-third  interest  in  the  land;  the  offer  being  accepted,  the  land 
was  surveyed  and  divided,  each  taking  for  his  exclusive  use  one-third  of  it.  How 
should  Harrison  and  Morton  divide  the  88000  received  from  Cleveland? 

8.  Seaman  and  Sullivan  entered  into  partnership  with  a  joint  capital  of 
$35500,  of  which  Seaman  invested  $22000.  During  the  existence  of  the  part- 
nership, each  withdrew  $1500,  and  it  was  agreed  that  no  interest  account  should 
be  kept,  and  that  Seaman  should  receive  f  of  the  gains,  and  sustain  the  same 
share  of  the  losses,  if  any;  while  Sullivan  should  receive  f  of  the  gains,  and  Sustain 
that  share  of  the  losses,  if  any.  At  the  time  of  the  dissolution,  the  resources  and 
liabilities  were  as  follows: 


Eesources. 
Cash $  2050 

Accounts  receivable 15850 

Real  estate 8100 


Liabilities. 

Notes  outstanding $21500 

Accounts  outstanding $16500 

Insurance  and  interest  due 2000 


Find  the  net  loss  of  the  firm,  and  each  partner's  net  insolvency  at  closing. 


2G0  EXAMPLES   IN"    PARTNERSHIP. 

Operation  and  Expijlnation. 


Total  liabilities $40000. 00 

Total  resources 2G000.00 

Net  insolvency 814000.00 

Seaman's  f  of  net  loss 129062.50 

Sullivan's  f  of  net  loss 17437.50 

Total  loss $46500. 00 

Proof. 

Seaman's  net  insolvency $8562.50 

Sullivan's  net  insolvency 5437.50 

Net  insolvency  of  firm $14000.00 


Seaman's  investment $22000 

Seaman's  withdrawal 1500 

Seaman's  net  investment $20500 

Whole  investment $35500 

Seaman's  investment 22000 

Sullivan's  investment $13500 

Sullivan's  Avithdrawal 1500 

Sullivan's  net  investment $12000 

Seaman's  net  investment .$20500 

Sullivan's  net  investment 12000 

Firm's  net  investment $32500 

Firm's  insolvency 14000 

Firm's  net  loss $46500 

Seaman's  |-  of  loss,  $29002.50,  less  liis  net  investment,  $20500  =  $8562.50, 
Seaman's  net  insolvency. 

Sullivan's  f  of  loss,  $17437.50,  less  his  net  investment,  $12000  =  $5437.50, 
Sullivan's  net  insolvency. 

S42.  To  Divide  the  Gain  or  Loss,  according  to  the  Amount  of  Capital  Invested, 
and  Time  it  is  Employed. 

Example. — A,  B,  and  C  are  partners  in  business;  A  invested  $3000  for  four 
years,  B  invested  $5000  for  three  years,  and  C  invested  $4500  for  two  years. 
How  should  a  gain  of  $15000  be  divided? 

Operation  and  Explanation. 

A's  investment  of  $3000  for  4  yr.  =  an  investment  of  83000  X  4,  or  $12000,  for  1  yr. 
B's  investment  of  $5000  for  3  yr.  =  an  investment  of  $5000  x  3,  or  $15000,  for  1  yr. 
O's  investment  of  $4500  for  2  yr.  =  an  investment  of  $4500  x  2,  or  $9000,  for  1  yr. 


A's  investment  for  1  vr. 

=  $12000 

B's  investment  for  1  yr. 

=  $15000 

C's  investment  for  1  yr. 

=      9000 

Total  investment  for  1  yr. 

=  $36000 

J0000 

:  1?000  : 

,  1^000  :  A'  gain. 

t 

5000 

$5000  =  A 

's  part  of 

gain. 

$0000 

:  1W0  : 

:  10000  :  B's  gain. 

n 

5 

1250 

5  X  1250  =  $6250  =  B's 

gain. 

?0000 

:  1^000  : 

:  0000  :  C's  gain. 

n 

5 

750 

5  X  750  =  $3 

r50  =  C's 

gain. 

EXAMPLES    IX    PAKTNEBSHIP.  267 

Rem AKK.— Should  withdrawals  of  capital  be  made  at  different  times,  or  additional  invest- 
ments be  made,  follow  the  steps  taken  above:  i.  e.,  by  subtracting  from  the  whole  investment 
for  1  year  (or  1  month)  the  Avhole  withdrawal  for  1  year  (or  1  month). 


BXAMPI.ES   FOR  PRACTICE. 

842.  1.  Three  persons  traded  together  and  gained  $900;  A  had  invested  in 
the  business  $1000,  for  C  months;  B  had  invested  $T50,  for  10  months;  and  C 
had  invested  §1200,  for  5  montlis.     How  should  the  gain  be  divided? 

2.  A,  B,  and  C  were  partners;  A  had  $800  in  the  business  for  1  year,  B  had 
f  1000  in  for  9  months,  and  C  had  $2000  in  for  8  months.  How  should  a  gain  of 
$2150  be  divided  ? 

3.  Martin  and  Eaton  were  partners  one  year,  Martin  investing  at  first  $5000, 
and  Eaton  $3000;  after  six  months  Martin  drew  out  83000,  and  Eaton  invested 
$1500;  they  gained  $3600.  What  was  the  gain  of  each,  and  the  present  worth 
of  each,  at  the  time  of  the  dissohition  of  the  partnership? 

4.  A,  B,  and  C  hired  a  pasture  for  6  months  for  $95.10;  A  put  in  75  sheep, 
and  2  months  hxter  took  out  40;  B  jiut  in  60  sheep,  and  at  the  end  of  3  months 
l^ut  in  45  more;  C  put  in  200,  and  after  4  months  took  them  out.  What  part 
of  the  rent  should  each  pay? 

5.  A,  B,  and  C  were  partners,  with  a  joint  capital  of  $18600;  A's  capital 
was  invested  for  6  months,  B's  for  10  months,  and  C's  for  1  year;  A's  part  of 
the  gain  was  $1260,  B's  $1500,  and  C's  $1200.  Find  how  much  Avas  invested  by 
each. 

6.  A  and  B  engaged  in  the  grocery  business  for  3  years,  from  March  1,  1885; 
on  that  date  each  invested  $1600;  June  1,  A  increased  his  investment  $400,  and 
B  drew  out  $300;  Jan.  1,  1886,  each  withdrew  $1000;  Jan.  1,  1887,  each  invested 
$1500.  How  should  a  gain  of  $7500  be  divided  at  the  time  of  the  expiration  of 
the  partnership  contract? 

7.  A  commenced  digging  a  ditch,  and  after  working  6  days  was  joined  by 
B,  after  which  the  two  worked  together  9  days,  when  they  were  joined  by  C. 
The  three  then  worked  12  days,  at  the  end  of  which  time  A  left  the  job  and  D 
worked  with  the  other  two  3  days  and  the  work  was  comoleted.  If  $92  was  paid 
for  the  work,  how  much  should  each  receive  ? 

8.  July  1,  1885,  A  and  B  commenced  business  with  a  capital  of  $7500,  fur 
which  A  furnished  -|  and  B  the  remainder;  May  1,  1886,  B  invested  $1500,  and 
A  withdrew  $600;  Oct.  1,  1886,  they  admitted  C  as  a  partner,  with  an  investment 
of  $4500;  Jan.  1,  1887,  each  partner  invested  $1000,  and  on  Jan.  1,  1888,  each 
partner  withdrew  $500.  On  closing  business,  Oct.  1,  1888,  it  is  found  that  a  net 
loss  of  $3000  has  been  sustained.     Find  each  partner's  proportion  of  the  loss. 

■9.  Olsen  and  Thompson  dissolved  a  three-year's  partnership  Aug  1,  1888, 
having  resources  of  $16500,  and  liabilities  of  $2150.  At  first  Olsen  invested 
$2750,  and  Thompson  $2500;  at  the  end  of  tlie  first  year  Olsen  drew  out  $1500, 
and  Thompson  invested  $3000;  six  months  later  each  invested  $1200.  Xo 
interest  account  being  kept,  what  has  been  tlie  gain  or  loss,  and  the  share  of 
each  partner,  if  apportioned  according  to  average  investments  ? 


268  EXAMPLES    IN"    PARTNERSHIP. 

10.  Simmons  and  Sawyer  commenced  business  with  $25500  capital,  of  which 
Simmons  invested  ^13500.  It  was  agreed  that  Sawyer  sliould  liave  $1200  a  year 
sahiry  for  attending  to  tlie  business,  and  that  tlie  net  gain  should  be  divided  in 
proportion  to  investments.  At  the  close  of  1  year  the  partnership  was  dissolved, 
the  firm  having  resources  to  the  amount  of  $;}7500,  and  liabilities,  otlier  than  for 
Sawyer's  salary,  to  the  amount  of  $4150.  If  neither  made  witlulrawals  during 
the  year,  what  was  the  interest  of  each  partner  at  closing? 

11.  Drew,  Allen,  and  Brackett,  each  invested  1^15500  in  a  business  that  gave 
the  firm  a  profit  of  $21000  in  one  year.  Nine  months  before  dissolution,  Drew 
increased  his  investment  $3000,  and  Allen  and  Brackett  each  Avithdrew  $3000; 
six  months  before  dissolution,  Allen  invested  $2000,  and  Drew  and  Brackett  each 
drew  out  $2000;  three  mouths  before  dissolution,  Brackett  invested  $1000,  and 
Drew  and  Allen  each  drew  out  $1000.  If  no  interest  account  was  kept,  and  the 
gain  be  divided  according  to  average  investment,  what  is  each  partner's  share  ? 

12.  A  and  B  formed  a  copartnership  for  3  years,  A  investing  $7200,  and  B 
investing  $5400.  At  the  end  of  G  months  A  increased  his  investment  by  $1500, 
and  B  Avithdrew  $900;  one  year  before  the  expiration  of  the  partnership,  each 
withdrew  $1000;  and  6  months  later  each  invested  $500.  The  net  loss  was 
$2400.  How  much  should  be  sustained  by  each,  if  sustained  according  to  aver- 
age investment;  and  if  each  be  credited  for  interest  at  ^^  on  investments  and 
be  rliarged  interest  on  withdrawals,  what  will  be  the  present  worth  of  each  at 
closing  ? 

13.  Sept.  1,  1883,  Martin  and  Gould  engaged  in  partnership  for  5  years,  Martin 
investing  $13000,  and  the  firm  assuming  his  debts,  amounting  to  $2750;  Gould 
investing  $9G00,  and  the  firm  assuming  his  debts,  to  the  amount  of  $1050.  At 
the  end  of  the  first  year  Martin  withdrew  $2000,  and  Gould  invested  $800.  At 
the  end  of  the  second  year  Cole  was  admitted  as  an  equal  partner,  he  making  au 
investment  of  $C000.  One  year  later  each  drew  out  $1000;  and  six  months 
before  the  partnership  contract  expired,  each  invested  $2500.  Sept.  1,  1888, 
the  partnership  was  dissolved,  at  which  time  it  was  found  that  a  net  loss  of  $7500 
has  been  sustained.  If  the  loss  was  shared  in  proportion  to  average  investment* 
what  was  the  loss  o^  each  partner? 

MISCELI-ANEOUS   EXAMPLES. 

1.  Hart,  of  Kansas,  and  Brown,  of  New  York,  form  a  copartnership  in 
the  grain  business;  Hart  to  make  jjurchases.  Brown  to  effect  sales,  and  they 
agree  to  share  equally  the  gains  or  losses.  Brown  sent  Hart  $12,000  cash;  Hart 
bought  grain  to  the  value  of  $14,382.50,  and  sent  Brown  40  car  loads  of  corn,  of 
600  bushels  each,  which  Brown  sold  at  65^  per  bushel.  Hart  paid  traveling 
expenses  to  the  amount  of  $438.20,  and  Brown  paid  freight  $1249.70.  At  the 
close  of  the  season  Hart  had  in  his  possession  wheat  to  the  value  of  $1128.42, 
and  Brown  had  on  hand  8300  bushels  of  oats,  worth  28^  per  bushel  in  the  New 
York  market.  They  then  dissolved  partnership,  each  taking  the  grain  in  his 
possession  at  the  values  stated.  What  has  been  the  gain  or  loss,  and  how  should 
the  partners  settle  ? 

Remark. — By  application  of  the  principlea  of  debit  and  credit,  as  used  in  book-keeping,  a 
book-keeper  may  with  ease  and  certainty  close  np  the  affairs  of  a  partnership  involving  any 
agreed  division  of  gains  or  losses,  interest  conditions,  or  those  of  prior  or  subsequent  insolvency. 


EXAMPLES   IX    PARTNERSHIP. 


269 


Dr, 


Hart, 


Or.      Br 


!|12000.00 
1128.42 

$13128.42 
3183.29 


$16311.71 


114382.50 

438.20 

1491.01 


$16311.71 


Operation. 
Brown. 


Cr. 


$15600.00 
2324.00 


$17924.00 


$12000.00 
1249.70 
1491.01 


$14740.71 
3183.29 

$17924.09 


Dr. 


Grain. 


Cr. 


$14382.50 

438.20 

1249.70 


$16070.40 
1491.01 
1491.01 


$19052.42 


$17924.00,  Brown's  debit. 
14740.71,  Browirs  credit. 


$15600.00 
1128.42 
2324.00 


$19052.42 


$19052.42,  sales  of  grain. 
10070.40,  purchases  of  grain. 


$3183.29,  excess  received  by  Brown,  or 
the  amount  due  from  Brown  to  Hart. 


2  )  2982.02,  net  gain  of  firm. 


1491.01,  net  gain  of  each. 

ExPivANATiON.— Credit  Brown  for  the  $12000  cash  sent  by  him  to  Hart,  and  debit  Hart  for 
the  same  amount.  Credit  Hart  for  the  $14382.50  paid  by  him  for  grain,  and  debit  Grain  for 
the  same  amount.  Credit  Grain  for  $15600,  the  price  received  by  Brown  for  the  40  car  loads 
of  corn,  and  debit  Brown  for  the  same  amount.  Credit  Hart  for  the  $438.20  expenses  paid 
by  him,  and  debit  Grain  for  the  same  amount,  as  an  element  of  its  cost.  Credit  Brown  for  the 
$1,249.70  freight  paid,  and  debit  Grain  for  the  same  amount  as  an  added  element  of  its  cost. 
Now  under  the  dissolution  agreement,  debit  Hart  for  $1128.42,  the  inventory  value  of  the 
grain  taken  by  him,  and  credit  Grain  for  the  same  amount,  as  having  virtually  been  sold  to  Hart. 
Debit  Brown  for  $2324,  the  inventory  value  of  the  oats  taken  by  him,  and  credit  Grain  for 
that  amount,  as  having  virtually  been  sold  to  Brown.  Having  now  disposed  of  all  the  grain, 
the  difference  between  its  cost,  Dr.,  and  the  returns  from  its  sales,  Cr.,  will  show  the  gain  or  loss. 
Foot  the  debits,  and  find  the  total  cost  to  have  been  $16070.40;  foot  the  credits,  and  find  the 
total  receipts  from  sales  to  have  been  $19052.42,  showing  a  net  gain  of  the  difference,  or 
$2982.02,  one-half  of  which,  or  $1491.01,  should  go  to  the  credit  of  each  partner.  Debit 
Grain  for  Hart's  one-half  of  the  gain,  $1491.01,  and  credit  Hart  for  the  same  amount,  to  which 
he  is  entitled  by  the  partnership  agreement;  and  for  like  reasons,  debit  Grain  for  $1491.01,  as 
Brown's  one-half  of  the  gain,  and  credit  Brown  for  the  same  amount,  as  his  oae-half  of  the 
gain,  and  find  that  while  Brown  is  entitled,  as  shown  by  his  credits,  to  only  $14740.71,  he 
has  actually  received,  as  shown  by  his  debits,  $17924,  or  that  he  has  received  the  difference 
$3,183.29,  more  than  he  is  entitled  to  receive.  Also  find  that  while  Hart  is  entitled,  as  shown 
by  his  credits,  to  receive  $16311.71,  he  has  actually  received,  as  shown  bj-^  his  debits,  only 
$13128.42,  or  that  he  has  received  the  difference,  $3183.29,  less  than  is  due  him.  If  then. 
Brown  pays  the  excess,  $3183.29,  that  he  has  received,  over  to  Hart,  the  accounts  of  both,  as 
well  as  the  Grain  account,  will  be  in  balance,  and  the  obtained  results  will  be  shown  as  follows: 

1st.  Net  gain,  $2982.02.     2d.  Net  gain  of  each,  $1491.01.     3d.  Brown  owes  Hart  $3183.29. 

2.  Hopkins  and  Hawley  formed  a  partnersliip  Sept.  1,  1880,  for  two  years,  and 
agreed  that  the  gains  or  losses  in  the  business  should,  on  settlement,  be  adjusted 
according  to  the  average  investment.  Sept.  1, 1886,  Hopkins  invested  $0250,  and 
Hawley  invested  $4500.  Three  months  later  each  invested  $1750.  On  Mar.  1, 
1888,  Hopkins  drew  out  $3000,  and  Hawley  invested  $2000.  How  should  a  gain 
of  $9400  be  divided  ? 

S.  Three  boys  bought  a  watermelon  for  24'/,  of  which  price  Charles  paid 
9^,  John  8^  and  Walter  7?^.  Ralph  offered  24^  for  one-quarter  of  the  melon, 
which  offer  was  accepted  and  the  melon  divided.  How  should  tlie  24j^  received 
from  Ealph  be  divided  among  the  other  three  boys? 


270  EXAMPLES   IN    PARTNERSHIP. 

Jf..  At  the  timo  of  closing  business,  the  resources  of  a  firm  were:  Cash, 
$931.50;  Mdsc,  per  inventory,  S13196.25;  notes  and  accounts  due  it,  $8154; 
interest  on  same,  $211.50;  real  estate,  $11150.  Tlie  firm  owed,  on  its  notes, 
acceptances  and  bills  outstanding,  $7142,  and  interest  on  the  same,  $348.50;  and 
there  was  an  unpaid  mortgage  on  tlie  real  estate  of  $2500,  with  interest  accrued 
thereon  of  $88.50.  If  the  invested  capital  was  $22500,  what  was  the  net  solvency 
or  net  insolvency  of  the  firm  at  closing,  and  how  much  has  been  the  net  gain  or 
net  loss  ? 

5.  Gray,  Snyder  and  Dillon  entered  into  partnership  with  equal  investments, 
and  agreed  that,  in  case  no  withdrawals  of  capital  were  ma4e,  and  no  added 
investments  made  by  either,  they  should  share  the  gains  or  losses  equally;  but  in 
case  either  party  increased  or  diminished  his  investment,  the  gains  or  losses 
should  be  shared  according  to  average  investment.  At  the  end  of  G  months  Gray 
withdrew  $2000,  and  Snyder  $3000,  and  Dillou  invested  $5000.  Three  months 
later  Gray  invested  $1000,  and  Snyder  and  Dillon  each  withdrew  $1500.  At  the 
end  of  the  year  they  dissolved  the  partnershi]),  having  as  total  resources,  $51000; 
total  liabilities,  $10500.  No  interest  account  having  been  kept,  what  was  the 
present  worth  of  each  at  closing,  and  what  was  the  gain  of  each,  the  whole  gain 
being  $6900  ? 

6.  Phelps,  Kogers,  and  Wilder  enter  into  partnership  for  five  years.  Phelj)S 
invested  $10000;  Rogers,  $20000;  and  ^Yilder,  $30000.  At  the  end  of  each  year 
Phelps  withdrew  $1000;  Rogers,  $1000;  and  Wilder,  $1800.  Upon  final  settle- 
ment, tlie  value  of  the  jmrtnership  property  was  $57200.  How  much  of  this 
sum  should  each  receive? 

7.  Apr.  1,  1884,  Smith  and  Jones  commenced  business  as  partners,  Smith 
investing  $8000,  and  Jones  $6000;  six  months  later  each  increased  his  investment 
$1500;  and  on  Jan.  1, 1885,  Brown  was  admitted  as  a  partner  with  an  investment 
of  $2400.  On  Oct.  1,  1885,  each  partner  drew  out  $1500;  on  K\)V.  1,  1886,  Smith 
and  Jones  each  drew  out  $1000,  and  Brown  invested  $6000.  On  Jan.  1,  1889, 
it  was  found  that  a  net  gain  of  $37500  has  been  realized.  What  was  the  share  of 
each?  If  by  agreement  Smith,  at  final  settlement,  was  to  be  allowed  $1200  per 
year  for  keeping  the  books  of  the  concern,  what  was  the  present  worth  of  each  ? 

8.  Burke,  Brace,  and  Baldwin  became  partners,  each  investing  $15000,  and 
each  to  have  one-third  of  the  gains  or  sustain  one-third  of  the  losses.  Burke 
withdrew  $2100  during  the  time  of  the  partnership,  Brace  $1800.  and  Baldwin 
$2000.  At  close  of  business  their  resources  were:  Cash,  $3540;  Mdse.,  114785; 
notes,  acceptances,  and  accounts  receivable,  exclusive  of  partner's  accounts, 
$16250;  real  estate,  $28500.  They  owed  on  their  outstanding  notes  $8125,  and 
on  sundry  personal  accounts  $1950.  Find  the  present  worth  of  each  partner  at 
closing. 

9.  Parsons  and  Briggs  became  partners  Apr.  1,  1887,  under  an  agreement 
that  each  should  be  allowed  G^  sim])le  interest  on  all  investments,  and  that,  on 
final  settlement,  Briggs  should  be  allowed  10;^  of  the  net  gains,  before  other 
division,  for  superintending  the  business,  but  that  otherwise  the  gains  and  losses 
be  divided  in  proportion  to  average  investment.  Apr.  1,  1887,  Parsons  invested 
$18000,  and  Briggs  $4000;  Jan.  1,  1888,  Parsons  withdrew  $5000,  and  Briggs 


EXAMPLES    IN    PARTNERSHIP.  5J71 

invested  $3000;  Aug.  1,  1888,  Briggs  withdrew  I^ISOO;  Dee.  1,  1888,  tlie  ])artners 
agreed  upon  a  dissolution  of  the  partnership,  having  resources  and  liabilities  as 
follows: 


Liabilities. 

Notes  and  acceptances $6520.00 

Outstanding  accounts 21246.50 

Kent  due 1200.00 


Resources. 

Cash  on  hand  and  in  bank $  1101.05 

Accounts  receivable ]  6405. 50 

Bills  receivable 2550.00 

Int.  accumulated  on  same 287.41 

Mdse.  per  inventory. —     9716.55 

If,  of  the  accounts  receivable,  only  80^  prove  collectible,  what  has  been  the  net 
gain  or  loss?  What  has  been  the  gain  or  loss  of  each  partner?  What  is  the  firm's 
net  insolvency  at  dissolution?    What  is  the  net  insolvency  of  each? 

10.  Bradley  and  Maben  became  partners  July  1, 1885,  under  a  3-year's  contract 
which  provided  that  Bradley  should  have  $1500  each  year  for  superintending 
sales,  and  that  Maben  sliould  have  $1000  each  year  for  keeping  the  books  of  the 
concern,  and  that  these  salaries  should  be  adjusted  at  the  end  of  each  year  and 
before  other  apportionment  of  gains  or  losses  was  made.  July  1,  1885  each 
invested  $12500.  Six  months  later  each  increased  his  investment  $5000.  July 
1,  1S86,  Bradley  drew  out  $3600,  and  Maben  drew  out  $3000.  Oct.  1,  1886, 
Bradley  withdrew  $1000  and  Maben  invested  $2000.  July  1,  1887,  each  drew 
out  S1500.  At  the  expiration  of  the  time  of  the  contract  the  resources  exceeded 
all  liabilities  $47280.  What  was  the  gain  of  each,  and  the  present  wortii  of 
each  ? 

11.  Clark,  Wilkin  and  Ames  bought  a  section  of  Kansas  land  for  $6400,  of 
which  Clark  paid  $1600,  Wilkin  $2000,  and  Ames  the  remainder.  Wheeler 
offered  $4000  for  one-fourth  of  the  land;  the  offer  was  accepted,  and  each  of  the 
four  had  set  apart  a  quarter-section  for  his  exclusive  use.  How  shall  the  money 
received  from  Wheeler  be  divided  ? 

12.  A,  B,  and  C,  formed  a  copartnership  for  2  years,  investing  equal  sums, 
with  the  agreement  that  each  shall  receive  interest  at  the  rate  of  G^  on  all  sums 
invested,  be  charged  interest  at  the  same  rate  on  all  sums  withdrawn,  and  the 
gains  or  losses  shown  on  final  settlement  be  apportioned  according  to  average  net 
investment.  Three  months  after  the  formation  of  the  partnership  A  drew  out 
$1200,  and  six  months  later  B  and  C  each  drew  out  $1000,  and  A  invested  $6000; 
at  the  end  of  the  first  year  each  drew  out  $500.  On  closing  the  affairs  of  the 
firm,  the  following  statement  was  made:  net  gain,  $15000;  present  worth,  $75000. 
What  was  the  original  investment  of  each?  What  was  the  present  worth  of  each 
at  the  time  of  dissolution?     What  Avas  each  partner's  share  of  the  gain? 

13.  A  and  B  became  partners  for  one  year;  A  investing  f  of  the  capital,  and 
B  f ;  the  agreement  being  that  the  gains  or  losses  shall  be  apportioned  accord- 
ing to  average  net  investment,  and  that  each  partner  be  allowed  6ffe  interest 
per  annum  on  all  investments,  and  be  charged  interest  at  that  rate  on  all 
sums  withdrawn.  At  the  end  of  the  year  the  firm  had  as  resources:  Mdse., 
per  inventory,  $21460;  real  estate,  $15000;  casli,  $1950;  bills  receivable, 
$13146.50;  interest  accrued  on  the  same,  $519.25;  accounts  due  it,  $11218.50; 


272  EXAMPLES   IN    PARTNERSHIP. 

store  furniture,  $1320;  delivery  wagons  and  horses,  12100.  The  liabilities  were: 
mortgage  on  real  estate,  $7000;  interest  on  same  accrued,  $210;  notes  outstand- 
ing S26950;  interest  accrued. on  same,  1811.75.  The  firm  owes  Barnes,  Clay  & 
Co.,  of  Boston,  $33560.  It  is  found  that  33 J  per  cent,  of  the  accounts  due  the 
firm  are  uncollectible.  If  the  firm's  losses  during  the  year  have  been  $12000,  how 
much  was  invested  by  each  partner  ?  What  is  the  present  worth  or  net  insolvency 
of  the  firm,  and  of  each  partner,  at  closing  ? 

14.     Clay  and  Hard  commenced  business  Nov.  1,  1883,  with  the  following 
resources: 

Clay  invested  cash $10000  |  Hard  invested  Mdse.,  valued  at ..$13500 

Store,  valued  at. ..-  12000  j  Cash 3000 

Marble  fixtures,  valued  at 1500  i  Good  will  of  trade,  valued  at. . .     7500 

The  firm  assumed  an  outstanding  mortgage  on  the  store  of  $6000,  and  a  note 
made  by  Hard  for  $3000,  and  due  without  interest  July  1,  1884.  Jan.  1,  1884, 
each  partner  withdrew  $300:  May  1 ,  1S86,  Clay  withdrew  $2000,  and  Hard  invested 
the  same  amount.  Jan.  1,  1887,  Dunn  was  admitted  to  the  partnership,  with  a 
cash  investment  of  $4500.  Xov.  1,  1887,  each  partner  invested  $1000;  and  on 
!N"ov.  1,  1888,  the  partners  agreed  upon  a  dissolution,  the  following  being  shown 
irom  the  ledger  of  the  firm: 


Liabilities. 

Xotes  and  acceptances $3825. 00 

Interest  on  notes 114.60 

Balance  of  mortgage  unpaid..  2150.00 

Taxes  on  store,  due 75. 40 

Due  Hard  for  keeping  the  books  5000.00 


Besources. 

Mdse. ,  per  inventory $48450. 50 

Cash '- 10918.20 

Accounts  receivable 23416.80 

Eeal  estate 15000.00 

Movable  fixtures  and  sundries,     3114. 50 

It  was  agreed  that  Hard  should,  at  the  time  of  dissolution,  be  allowed  $1000 
per  year  for  keeping  the  books  of  the  concern.  If  no  interest  account  was  kept 
and  the  gains  or  losses  be  apportioned  according  to  average  investment,  what  are 
the  net  resources  of  the  firm  at  closing  ?  What  has  been  the  net  gain  or  loss  ? 
What  has  been  the  gain  or  loss  of  each  parter  ?  What  is  the  present  worth  of 
each  at  closing  ? 


ANSWERS 


Page  12. 

Art.  64. 

1.  45. 

.2.  306. 

S.  217. 

4.  1647. 

5.  979. 

6.  262. 

7.  853. 
5.  599. 
9.  1053. 

iO.  1610. 

Art.  65. 

1.     3342. 
.^.     22512. 

26052. 

161840. 

223732. 

2967515. 
7.     813496. 
S.     21423493. 
9.     24543879. 
10.     8179519. 


1. 
S. 
3. 

4. 

5. 

6. 

7. 

S. 

9. 
10. 
11. 
12. 
13. 


Art.  66. 

133.36. 

530.80. 

553.61. 

629.23. 

421.34. 

536.91. 

948.69. 

91.30. 

314.61. 

296.19. 

488.35. 

260.'54. 

473.43. 

Art.  67. 

$3102. 

Page  13. 

5530  pounds. 
33200  feet. 
5114836332. 
18 


5. 

6457434373. 

Page  22. 

(>. 

515. 

Art.  91. 

7. 

599100. 

1. 

126. 

8. 

£919760700. 

2. 

124. 

9. 

$519949564.38. 

3. 

54. 

10. 

£87197000. 

4. 

300. 

11. 

168  in. 

6. 

204. 

12. 

$178586. 

6. 

450. 

7. 

182. 

Page  14. 

8. 

87. 

13. 

513281. 

9. 

114. 

u. 

50291783. 

10. 

475. 

15. 

3501409. 

11. 

408. 

12. 

4088. 

Page  15. 

13. 

750. 

16. 

$3361127356. 

u. 

680. 

15. 

1248. 

Page  18. 

16. 

693. 

Art.  80. 

17. 

1197. 

18. 

832. 

1. 

613. 

19. 

3330. 

2. 

1609. 

20. 

712. 

3. 

2022. 

21. 

1440. 

4. 

13890. 

22. 

572. 

5. 

50000. 

23. 

585. 

G. 

64365. 

24. 

3015. 

/  , 

151223. 

25. 

1300. 

<S'. 

57006. 

26. 

6987. 

.9. 

1407503. 

27. 

11184. 

10. 

213305. 

28. 

817. 

11. 

449. 

29. 

2553. 

12. 

30889825. 

30. 

4554. 

13. 

790000. 

31. 

7735. 

u. 

500. 

32. 

1540. 

33. 

2250. 

Page  19. 

34. 

3450. 

15. 

3175  bushels. 

35. 

1298. 

16. 

139886  feet. 

36. 

13590. 

17. 

1594. 

37. 

8550. 

18. 

5  and  38576  rem. 

38. 

5250. 

19. 

929496. 

39. 

4500. 

20. 

1984  dollars. 

40. 

8679. 

21. 

12960  acres. 

22. 

$53440. 

Art.  9:j. 

23. 

$6250. 

1. 

1608. 

2J,. 

$140. 

2. 

2535. 

25. 

708  miles. 

3. 

12012. 

4. 

1866. 

5. 

7245. 

6. 

9624. 

7. 

2650. 

8. 

23188. 

9. 

7665. 

10. 

3516. 

11. 

23413. 

12. 

9576. 

13. 

12976. 

14. 

14427. 

15. 

42084. 

16. 

1743. 

17. 

4152. 

18. 

12342. 

19. 

8333. 

20. 

35610. 

21. 

7872. 

22. 

27120. 

23. 

6454. 

24. 

53130. 

Page  23. 

Art.  94. 

1. 

19512. 

2. 

496736. 

3. 

7188. 

4- 

28210. 

5 

559790. 

6. 

6410556. 

7 

17180824. 

8. 

229291455. 

9. 

6605212120. 

10. 

89336820048. 

11. 

1486262400360. 

12. 

2651701850220. 

13. 

463437665439. 

14- 

10768229616048. 

15. 

321453090615. 

Page  24. 

Art.  99. 

1. 

615. 

0 

357. 

3. 

2664. 

4. 

41652. 

5. 

90855. 

274 


ANSWERS. 


6. 

8352192. 

7. 

7809840. 

S. 

7809840. 

9. 

7809840. 

10. 

7809840. 

11. 

7809840. 

12. 

7809-40. 

13. 

5184. 

u. 

5184. 

15. 

5184. 

Page  25. 

Art.  100. 

1. 

10017000. 

2. 

18941400. 

3. 

106326. 

4. 

257322000. 

5. 

41325000. 

G. 

252000. 

tsi 

864450. 

8. 

46232353. 

9. 

145152. 

10. 

109515. 

11. 

1305  cents. 

J2. 

990  dollars. 

IS. 

3030  dollars. 

u. 

2700  dollars. 

15. 

57708. 

16. 

238800800. 

17. 

2285  dollars. 

18. 

15378  dollars 

19. 

$645  gained. 

no. 

3720  dollars. 

Page  26. 

Art.  101. 

$385560000. 

$189739175. 

$123224. 

$15147.50. 

$116816. 

26784  ft. 

7.  6717  $?. 

8.  $200. 

9.  1689^. 

10.  3361  {J. 

11.  $844. 

12.  92700  pairs. 

Page  27. 

15.  358302^. 
H.     2915  lb. 

16.  71700. 


16.  $8158  gained. 

in.  $675. 

IS.  $2230. 

19.  $1899600. 

20.  $1649  gain. 


Page  29. 

Art.  112. 

1.  8,  4.  2. 

2.  10,  5,  4,  2. 

3.  14,  7,  28,  8,  4. 

Jf.  18,  30,  15,  6,  10. 

5.  5,  3,  9,  15. 

6.  9,  2,  3,  18,  4. 

7.  12,  36.  6,  3. 

5.  12,  21,  42,  7,  4. 
9.  20.  4.  50,  25,  10. 

10.  4,  12,  6,  2,  3. 

11.  25,  5. 

12.  12,  4,  16,  8,  24. 

13.  8,  16,  2,  32,  4. 
U.  6,40,15,24,10. 
15.  20,  5,  8,  4,  10. 

I  16.  12,  18,  24.  36, 
48.  6. 

i7.  5,  25.  35. 

18.  16,  12,  3,  8,  6. 

19.  '36,  54,  12,  18, 

9,  4. 

20.  40,  20, 10. 25.  50. 

Page  30. 

Art.  117. 

1.  323. 

2.  315. 

3.  281. 

4.  529. 

o.  3945|. 

6.  6744|. 

7.  17?ii- 

5.  13023y\. 
SI.  6234L|. 

10.  417230xV 

11.  12870. 
if.  10880y''5. 
13.  26751 . 
i.^.  4637. 
15.  475. 

i6.  184GlyV- 

i7.  1361080^5. 

18.  56026^7. 

19.  11137tV 


20. 
21. 


S. 

9. 
10. 
11. 
12. 
13. 

u. 

15. 


8. 

0. 
10. 
11. 
12. 


111930yV 
706369^^. 

Page  31. 

Art.  119. 

3.  7.  217. 

20.  5.  217. 

11.  9.  217. 

217.  10.  45. 

217.  11.  45. 

217.  IJ.  45. 

Page  32. 

Art.  121. 
AjSl 

19H. 

26fi 

34^- 
179_3^. 

371^. 

371^. 

sum- 

371^. 
3T1A\- 

5173|f 

QOl 88  7 

11005X4. 
7965^.' " 

Page  33. 

Art.  124. 

1039 

i~4nj. 

026  4 
Kl  84 

60A"A. 


1  r;  S9T4 


Art.   125. 

$103055. 

IGOOO  acres. 

621  acres. 

$3. 

4yW  miles,  and 

$138i||. 
36666667. 
15353637-6  rem. 


8. 

9. 

10. 


11. 


29. 
30. 


10. 
11. 


13. 

6. 

715  acres. 

Page  34. 

2114. 


Page  35. 

Alt.   12T. 

1. 

36. 

2. 

15. 

3. 

5. 

4. 

12. 

5. 

50H- 

6. 

3796v«. 

y_ 

481/^. 

8. 

12|if. 

0. 

315lt|. 

10. 

2823H- 

11. 

746^<V. 

12. 

IIVA 

13. 

310|Hf. 

14. 

61286t85V5- 

15. 

265095iil§. 

16. 

3960||^f. 

17. 

11471  Hi. 

18. 

7QQ7  3  503 

19. 

244ISUI. 

20. 

1000,1?^,. 

21. 

18053i§iH- 

;   -, 

0043537  6 

23. 

18555|f|f. 

24. 

5067970if|f. 

25. 

160212if||. 

26. 

10000,^14^ 

800044^^5^. 

Art.  128. 

0205  1 113 
~7  3TXr4TS- 

$07605t4|l. 
4025. 

KJl 4  1 7  00 

$15416||§§. 
18/^  lb. 
$641025f,=^.   and 

$53418H§f 
14||f  miles. 
348409iVff       lb. 
copper,  and  104- 

070jVij  lb.  tin. 
173/ii'V  miles. 
1720  bbl. 


Page  36. 

12.  164|^^  years. 

13.  $2906070732  i% 

and  $14378213- 
08/j. 
U.     257|§5  men. 

n.     $115009119aj  and 
$544888262fJ. 
18.     34r\. 

20.     86781  lim. 

32202S698 
0O         Ae 4  5  7  2  4  11 


Page  37 

Art.  i:U. 

1. 

46. 

2. 

82 1. 

3. 

69f. 

4. 

4Hf. 

5. 

47  miles. 

6. 

$13764^4. 

7. 

76^. 

Alt.  13.i. 

1. 

64. 

2. 

29. 

3. 

885. 

4. 

296. 

6. 

19. 

6. 

742. 

7. 

8751. 

8. 

8906. 

9. 

71237. 

10. 

17959. 

Page  39. 

Art.   l.jy. 

1.  3,  3,  and  3. 

2.  3,  3,  and  13. 

3.  3,  5,  and  11. 

4.  3,  and  31. 

5.  2,  2,  2,  3,  3,  3, 

and  11. 

6.  2,  3,  5,  5,  and  7. 

7.  2,2,2,2,3,  and  3. 

8.  5,  5,  5,  5,  5,  and  5. 

9.  2,  2,  2,  2,  2,  2,  3, 

3,  3,  and  13. 
10.     2,  3,  5,  7,  13.  17, 
and  19. 


ANSWERS 

375 

Page  40. 

Page  46. 

G. 

H.-4  7         9«e          26SS 
18»>     '18»>      "T8  8  > 

Art 

14G. 

Art.  173. 

V,V.and^«,V- 

1. 

11, 

7.     14. 

1, 

62                p          897 

7. 

3376      SOOO      7660 
?000<   »009>  ^SOV> 

,J. 

12. 

8.     50. 

2. 

8400     5800     anf\ 
9o00'  iroOS'  '»"" 

3. 

16. 

.9.     151. 

3. 

2260 
»(fOO- 

4- 

18. 

10.     63. 

4. 
5. 

If.    io.    «v,'A. 

S. 

V2T.^W.W(f-. 

5. 
G. 

52. 

45. 

11.  70. 

12.  25. 

3«an 
-T50  • 

Art.  174. 

0. 

1449000      708760 

»4b;ooo->  ^rsxim' 

Pag 

e41. 

1. 

h          6\      5^.5. 

BS31500            anl\ 
S4SOO0'.         *"<! 

Art 

148. 

2. 

¥•     7.    -%v-. 

7360000  ■ 
SI45  0US  > 

1. 

4. 

7.     7. 

3. 

-V.     <?•    HS^- 

75  8  000    1575000 
945U00'    SfBOOo 

24. 

8.     23. 

4- 

30«           q           20801 
-7      •        ■^-        --?5       • 

.13835000         j,n/1 

54  5  000^)    anu 

J. 

2. 

'J.     21. 

5. 

19  3        //I           14  5  2  7 

7  3  5  0  0  0 
94500<y- 

4- 
5. 
G. 

17. 
51. 
4. 

10.  131. 

11.  1. 

12.  25. 

Page  47. 

Art.  176. 

n>. 

54000                 69400 
7  4  25  0'              TT5?ff> 
47025            S5  145 
7rffS0>          'Ji^oO- 
19800            33000 
7T5o<J»          7  42  3  0, 

Page  43. 

Art.  15.5. 

1.  4S0. 

2.  4o6. 
■J.  1872. 

4.  840. 

5.  840. 
(.■.  9504. 
7.  7920. 
<S'.  840. 

9.  2520. 

Page  44. 

Art.  15«. 

1.  64f. 

.9  2  5 

^.  agy. 

?  161 

o*.  3  25- 

4.  120. 

5.  971. 
G.  90. 

7  8064^ 

6'.  35x\. 

5.  1080. 

10.  86f. 

ii.  45  bu. 

12.  5  bbl. 

iJ.  38  bales  and  528 

yards. 

14.  720  yd. 

10.  5J  pieces. 

17.  4h 

18.  14. 

19.  41»ibbl. 
i^y.  2g|  sections. 


6|.       7.     22|i 


rs- 


ft »       ,       . . 

"If-      •^-      ""TSa- 
85V-  ^0-     14|?&. 


9.   ssn^ 


Art.  178. 

3  A-  2 

2  7         85 

3-  '•       Y2- 


15- 
S 


rs- 


9. 
10. 


Page  48. 

Art.  180. 


10 


8  8 


18  7  10 

■SB-  '•  TT5- 

18  o  84 

^?-  ''•  IJS- 


20 


40 


-3f-  ^-       T9J- 

Art.  182. 


18285  unA 

15120 
245  7  0- 
3840        13800 
■2'50StT'     ^5080> 


?  2  0  »  0  • 

18      lis      14        7  6,5 
"20<  "2  0"'  "2  0'      50  • 

•*'«    and  ^A** 
'So  >  "■""    20  • 


10. 


11. 


stnri    1486000 

anu  -YUfso 
Page  49. 

Art.  184, 


I  5 
T20> 


815 
BO-f> 


4  8  10  0 

"i"2(7)         I'SO' 

and  -1V5. 

_6  6  8  8  8  8  11 

^ffT'  KOTi       BO- 

358  q„,1      8  4 

6  0f>  ^"11  50-f- 

49S  380       13860 

lS5ff'T9  8ff»    198  0 

880       4950     a,wl 

1886 
i»8  0- 
1080  1248  13  00 
"iS60'  lff«TJ'  1B«0> 
13  65  3  180  8  82  0 
156-U>  1560>  fStfO 
and      780 

18      10       6        5       3  0 
GO'  «U>  BO'  "b0'>  so, 
240     nnrlSOO 
eu  '  ^""  "80  • 
24     40      9  90      6  90 
90'   90,      90  >    "9«> 
186        810        anr\ 
90  »     "90",      *"^ 
90 

4830      8698      148.8 

i80">  "190  >  T^r5» 

103  5       .,,1,1     180 
"ISO",    '*"*^    180' 

il-f    >      84  '        8f    » 
735         272        onH 

420 
'8f  • 

1280      1880  1890 
2520'  252ff,  5Tr2D» 

2  0  18  2  10  0 

2  3S0>  2520' 

816  0  2205 

2520'  2320' 

??i"    and   2?5? 


18 


2'3-_- 
8160 

2520'  2320' 

2240  Q„,l  2  2  88 
8520'  '"•"'J  2520' 
i  4  8  8  15  8 

8ff'  18(J>  18  0' 
492  1440  18 
1»0'  "180"'  18d> 
630  2  89  6  niiH 
18  0>  18  0"'  •*"" 
48 

18  0- 
1800      17.82      1760 
1980,  "rtSff>  T980, 
2  7720  1»H0 


1  «  uo 

1980,  ..V 
27720 
198  0"! 
1710 
17S0> 
9865 
l98ff> 
81  780 
TSSO  • 


1760 

rise 

1  »  HO 
19  8  0' 
730  80 
"1980'" 
10698 
T5da  ' 


276 


ANSWERS. 


IS. 


4. 
6. 
6. 
7. 
8. 
9. 
10. 

1. 

2. 
3. 
A. 
s: 

6. 

7. 


10 


6  5  1680  8  0  6,0 
840  STOO  2500 
S  7  0  8  1800  880 


Art.  186. 


10. 


TiTff' 


1.     3.  G.     31|. 

S.     ^\.      7.     2|. 
3.     4.         5.     4. 


3lf- 
01  9 


^.     4tV.    -?^-     2i| 


Page  50. 


Art.  188. 


1.     4|. 


J.171  1 

4t5T5- 
Art.   190. 

01    9  89 

m09 
.5U- 


58U- 

105f|. 

329A. 

8.  715U. 

9.  709xVg. 
108f. 


Page  51. 

Art.  191. 

1.    'i'-h-       ~-  4i- 

:?.     3^-       S.  lU- 

5.     2HB-      5-  ^■ 

4.  ^n-    10.  4^. 

5.  2t|.      ii.  3|f. 

6.  2|§.      i^.  5^V- 


Art.  193. 

i. 

lOoV//^. 

;?. 

244fJ. 

3. 

COoJUI- 

4- 

1055^. 

5. 

42  H. 

€. 

412H. 

7. 

lOOH  acres 

8. 

1104IU  lb. 

2644xV  lb. 
5693^    bu.    and 
I3341H- 


Page  52. 

Art.  194. 

1.  |.  9.     ^. 

2.  \.        10.     2^. 

3.  1^.    11.     A- 

A.    |.      -?-'•    !!• 

5.  |.    13.  m 

6.  |.        14.     If. 

7.  H.     i5.    A- 

5.  i. 

Art.  195. 

1.     tV-        ^-     *• 

193         ;-?  9 

li     -Z-J-    4. 

A-    i4.    i. 
|.      ic.    2^15. 

Art.  197. 

1.  i        7.    M- 

2.  h  S-  ^• 
5.  i.  9.  If. 
^.  iV-  io.  f. 
5.  A.  -Zi.  ^• 


•5 

25- 


>sS- 


Art.  198. 

1.     i,.        9.     3|f. 
-     ^%.      10.     6f. 
h        11.     4^. 

A-  ^~'-  5U- 

5.  A-  i5.  «>5^. 

6.  i.  i4.  2^- 
StIj.  15.  ^. 
H-  ^<^-  A- 

Page  53. 

Art.  200. 

1.  4.  9.  9|. 

^.  2f  10.  12*. 

5.  9.  ii.  9i 

4.  17i.  i^.  14^^ 

5.  2i.  iJ.  5A- 

6.  3y«g.  i4.  8it. 

7.  7,^.  15.  99A. 
.•?.  S^j.  i6.  ITOi. 

Art.  201. 

i.     2i|. 


2.  \m- 

3.  30J. 

4.  20. 

5.  in. 


6.  36|f 

7.  63x||7r- 
5.     8f. 


9. 
10. 
11. 

12. 
13. 

14. 
15. 


150^. 

39^. 

198|Sf. 

e;ses48 

59if. 
87,«|. 
9691. 


16.     487i 


1. 

f.  • 

5. 

4. 

5. 

6. 


9. 
10. 
11. 
12. 
13. 

14- 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
•^2 
23. 
24. 


Art.  202. 

31 
91 
1  85 

lA- 
6A. 

^^21 
125^-5. 

«74  0  3 

4i6o- 

mi- 

323 
55- 

fi  4S1 

5931^. 
49tVit- 


47f5  acres. 
6fJ  dollars. 

Page  54. 

$23HJ.  gain. 
660HH. 

n4|&i. 

ft  51 

$34i. 
$12690H- 

^0  i  *>'i~5' 


Page  55. 
Art.  204. 

.1.    n.     9.   4i. 


2. 

f- 

10. 

1*. 

3. 

2i. 

11. 

a. 

4- 

If- 

12. 

9. 

5. 

5A. 

13. 

3i. 

6. 

2. 

14. 

14. 

rv 

3j. 

15. 

18. 

8. 

14. 

Ai-t 

.  205 

1. 

25. 

9. 

344. 

2. 

7A- 

10. 

600. 

3. 

44. 

11. 

m- 

4. 

30. 

12. 

297i 

5. 

124. 

13. 

265. 

6. 

26^. 

14. 

679. 

7. 

462| 

.  15. 

7. 

8. 

94i. 

16. 

68. 

Alt 

.  207 

. 

1. 

3|. 

9. 

3f. 

2. 

6i. 

10. 

6i. 

3. 

10. 

11. 

12. 

4- 

5^. 

12. 

m. 

6. 

35. 

13. 

10. 

6. 

2. 

14. 

16. 

7. 

15. 

15. 

36. 

8. 

6. 

Art 

.  208. 

1. 

46f. 

9. 

63. 

Z. 

49. 

10. 

427. 

s. 

33|. 

11. 

45. 

4- 

77. 

12. 

84. 

6. 

49i. 

13. 

65^. 

6. 

2U 

14. 

168t»c 

7. 

152. 

15. 

6972. 

8. 

3|. 

16. 

448. 

Page  56. 

Art.  210. 

r  4  O  25 

2.     i.       10.     |. 

5.  A.     11.     If. 

4.  if.      12.     -Nn- 

6.  A-  13.  m- 

6.  y%.         14-        f§J- 

7.  2tV.   -?5.     A- 
*.     hi     1(^-     i- 

Art.  211. 
1.      $|.       4.     A- 

5.  iA.    5.     A- 
5.     IH.     6.     ^. 

Page  57. 

7.     llf,  and  8^. 


ANSWERS. 


277 


s. 

9. 
10. 
11. 
VS. 

IS. 

u- 
1. 

2. 

S. 

4- 

B. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
U. 
IS. 
16. 
17. 
18. 

19. 
SO. 


|90,  and  $60. 

$1. 

$i. 

$i. 

lifV- 

n  gal- 
Art.  213. 

1  17S9 


■'tboo- 
1340713^. 
10667478x^5. 

80134846IH. 
18786149735tV- 


1160851. 

u- 

$87^. 

561^  acres.        / 

159^t  bbl. 
$6A. 

$35|U  gain- 
Page  58. 

Art,  214. 

^-    lo:  A. 


XT- 

A-    12-  ^v- 
A-    13.    If. 

A- 

Art.  215. 

8        q     751 

T55-        ''•       'BS- 

^I'oV    i^.    24^i- 

xfr.  i5.  34ii. 

4i.  i^.  20|i. 

3f?.  i5.  li^V- 

99|§.  i6.  531  J. 

Page  59. 

Art.  217. 

1.     39|.     ii.     12^|. 


^- 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
U. 
15. 
16. 
17. 
18. 
19. 
20. 


19f  12.  3f. 

31f.  IS.  24. 

51|.  U.  67i. 

888|.  15.  81. 

20.  16.  40. 

22|.  17.  300. 

67/^.  18.  14f. 

53^.  i9.  36. 

3|f.  20.  21. 

Art.  218. 

$72. 

5  shares. 
40  families. 
24  bu. 
5  da. 

$iH- 

3089^,  or  3090 
sacks. 

Page  60. 

Art.  220. 

2i.      10.     f. 

U-    ii.    H- 

f    i~'.  14. 

16.     U. 


T5 
Art.  231. 

1     6 

1^- 

76 
T7K- 

1196 

809 

10886 
ItSTB- 
19 

8600 
8Sd8- 

351. 


18  da. 
13. 

12i  da. 
1223/A  bu. 
8341  cords. 
184U  bu. 


23.     279i  miles. 
Page  61. 

Art.  223. 

1.     2||. 

q        71 1 
J.        (35. 

6'' ^4 


5. 
6'. 
7. 
5. 
9. 
10. 


11. 

12. 


13. 

U. 
15. 
16. 
17. 
IS. 
19. 
20. 
21. 


25. 
26. 


29. 


30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 

38. 
39. 
40. 
41- 


$14000. 

$6772^1. 

3bu. 

21b. 

$135000. 

323^y5,  sulphur, 

215^Vff  salt-  •  ""^ 
1615fi  char. 

$52500. 

$80  watch,  and 
$35  chain. 

Page  62. 

$5|. 

15  days. 

$19000. 

$67837i. 

51 +  ^ 

$2. 

$248,701. 

3iWij  J^iles. 

$5. 

$13200. 

336  trees. 

1001b. 

$67i. 

$6.  $15  and  $16. 

62i  years. 

S  $1190,  H  $476 

and  R  $544. 
90^/5  years. 

Page  63. 

$35  and  $40. 

105. 

6tbu. 

22*  da. 

J  .$475,  C  $38| 

40  ft. 

94^  ft. 

Cow    $30,    colt 

$94. 
2  p.  M. 
36  ft. 
405.. 
H  $216,  C  $324. 


42. 
43 
U 
45 

46 

4 

48. 

49. 
50. 
51. 
.52. 
53. 
54. 

55. 

56. 


58. 
59. 
60. 

61. 
62. 

63. 
64. 
65. 
66. 


68. 


69. 


70. 


71. 


30t  years. 

6|  da. 

5Ay  da. 

425  da. 

67^  da. 

S  $180,  B  $150. 

150  ft. 

Page  64. 

1 3 

15- 

176  rd. 
14|  min. 
Ben  7^,  John  \<f. 
28  bu.  andSObu. 
5^  min.  past  1 

o'clock. 
27t\  min.  of  7 

o'clock. 
\0\^  min.  of  10 

o'clock. 
5^  min.    of  11 

o'clock. 
60. 

52^^  loss. 
$1500,   $3000, 

$4500,  $6000. 
62yST  yards. 
A  $19.75,  and  B 

$15.80. 
15  hr* 
4  min. 
55  yr. 
A  "  $11tV^,   B 

$14,Vy  and  C 

$10rl5- 

Page  65. 

B42^,  S25|;  B 

$16|.  S  $28i. 
llfl^da; 

C  $32|f f , 

II  S25|ff, 

T  $22|if 

L  $193Vt- 
16f  da. ;  A,  $44- 
|^;andB,$30 

1  .IS 

$7350.  A,  $2650; 

B,  $2700;  C, 

$2000. 
H,216;M,  129f; 

and  B.  64|. 
81  da. 
49H- 


27i 

3 

ANSWERS 

• 

u. 

434. 

13. 

.000900. 

'     9. 

5S1S 

4. 

106  /,>.     35 

75. 

H,$33i;M,|55f, 

11 

.00000009. 

'   10. 

7807 

5. 

lil- 

and  B,  %\\\\. 

1.',. 

54054054.005405- 

11. 

2001 

6. 

ISH- 

76. 

A,  135|  da.;  B, 

0054. 

12. 

18081 

Art.  251. 

169^    da.;     C, 
188i    da.;    D, 

16. 

17. 

103.587. 
640.64. 

13. 

u. 

^'s  5  00  • 

4926 

1. 

2 

129.341. 

848.1816. 
1652.461772. 

67*  da. 

IS. 

26.04002. 

15. 

1000267 

3. 

77. 

AandB,75ida. ; 

19. 

9019.029039. 

16. 

Tf^TboojMf 

4- 

12638.517762. 

A  and  C,  78ff 

20. 

7.7. 

'17. 

5A. 

5. 

2002.55141194. 

da.;  A  and  D, 

21. 

870.01. 

18. 

13AV 

6. 

8688.0148502. 

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Page  74. 

and     D,     36JA 
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D,  llSAV/j 

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1798.9425  bu. 

Page  60. 

Art.  338. 

1. 

Art.  240. 

.25. 
.0106. 

1. 

2. 

Art.  246. 

.0625. 
.65. 

15. 
16. 
17. 

395.8125  yd. 

376. 

9262. 

1. 

.026. 
07 

3. 

.00256. 

3. 

.275. 

Art.  253. 

3. 

i. 
5. 
C. 

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.0016. 
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5.7. 

4. 
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1. 
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2.23985+. 
1.7912 
1.9^7703945. 

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2146.9003. 

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710.00243. 

9. 

10. 

56973.805. 
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9. 

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1384.4959234662 
2. 

9. 
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11. 
12. 
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500.05. 

45.046. 

1001.0100. 

1890.090. 

850.05. 

1000.10. 

Art.  239. 

11. 
12. 
IS. 

u. 

15. 
16. 
17. 

33254.81. 

.00001876. 

10.007. 

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15.0015. 

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219760.0801. 

11. 
12. 
13. 
14. 
15. 
16. 
17. 

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6. 

1. 

2. 

3. 
1 

5569.3518126587- 
42. 

Page  75. 

Art.  355. 

.412. 

.52977. 

.6863. 

5.5264. 

1.545648. 

54.2294. 

1. 

11.107. 

IS. 

.046700004. 

18. 

.028. 

2. 

3. 

15.0014. 
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19. 

2v. 

.00068001. 
1101.10011. 

19. 

20. 
21. 

22. 
23. 

u. 

25. 

.308. 

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.94. 

.226. 

.034375. 

.76. 

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4- 
5. 
6. 

4. 
5. 
6. 

/- 

1106.0012. 
1600.16. 

10000000.000010. 
3.65. 

1. 

Page  72. 

Art.  244. 

-A- 

8. 

9. 

10. 

754.6005. 
10000.0999. 
.3148. 
213.889625. 

Page  70. 

3. 

iTcrV- 

11. 

.810. 

S. 

25400.11. 

4. 

a  u  u 

Page  73. 

12. 

135.25740. 

9. 

21.0015015. 

5. 

u- 

Art.  249. 

Page  70. 

10. 

.0000018018. 

6. 

661 

1. 

2 

11. 

.500. 

7_ 

98  9 

2. 

f 

Art.  257. 

12. 

.00005. 

S. 

iVs- 

3. 

II- 

1. 

.546. 

ANSWERS. 


279 


S.  .01968. 

3.  1.26875. 

4.  39.9024. 

5.  23469.986904. 

6.  4625520.705. 

7.  1. 
<^.  9. 

9.  .625000. 

10.  .87000. 

11.  7231.98325125. 

12.  49. 

13.  .1. 

U.  275400116.25610 

02754. 

lo.  $20217.72. 

IG.  $536.88. 

17.  $937.04. 

18.  $336.33. 
Page  77. 

Art.  260. 

1.  .25. 

2.  305. 

3.  250. 

4.  .5. 

5.  .05. 

6.  50. 

7.  500. 

8.  4000. 
5.  2000. 

i(?.  .002. 

11.  .000025. 

12.  183100. 
IS.  5.5875. 
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i5.  50. 

1C>.     .00007. 

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18.     4000. 

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21.     25. 

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23.     .15. 

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2G.     .1. 
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10000000. 
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27.  .02. 
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10000000000. 

Pajfe  78. 

28.  64006464C4047.0- 

4U6404000064. 
2U.     250252502527.75- 
2750000025. 

30.  40000400000.044- 

800440044. 

31.  400044440440.00 

0000140004. 

32.  30030330000.003- 

60306. 

33.  150001650151.80- 

1650000015. 

Page  80. 

Art.  366. 

1.  1991.1198244. 

2.  .61032010. 
■  3.     .1625. 

/         «oi 

"f-       TOOoO- 

5.  .0038462. 

C.  2116.99454." 

7.  187.2996. 

8.  $7,766. 

9.  41.299781. 

10.  7.029956572371- 

000. 

11.  999999999.99999- 

9998. 

12.  .90. 

13.  572501.2525. 
U.     $1100.869. 


i.5. 

12711.8755*. 

IG. 

10.625. 

17. 

1011.6. 

IS. 

6.875  da. 

19. 

$274.  .58. 

20. 

$24.9331-f 

8. 

9. 
10. 
11. 
12. 


Page  84. 

Art.  387. 

ooo^-. 

11100^. 

$2.41. 

$10.44. 

$214.68. 

$18. 

$510. 

$98.76. 

100980. 

375i^. 

26530. 

157320. 


Art.  389. 

/.     $88.84. 

Page  85. 

2.  $144.89. 

3.  $4337.77. 
.'i.  $221.53. 

5.  $378.07. 

6.  $877.29. 

7.  384.51. 
cS'.  1088*  0. 
.'/.  4073^ «. 

10.  3065* 5. 

11.  8968*. 

12.  2094.15. 

Art.  291. 

1.  $8179.88. 

Page  86. 

2.  $4859.76. 

3.  S5669.60. 

4.  $29.52. 
o.     $839.26. 

.S33.73. 


i;_ 


Art.  393. 

1.     $2284.35. 

$1315.63. 

$7.13. 

950  bu. 

$344.73  gain. 

$91.10. 


Page  88. 

Art.  394. 

7 

26  and  39. 

8 

16^\  min.  j 

3  o'clock. 

9 

Midnight. 

10 

B.    12,     A. 

C.  78. 

11 

B.   840   and 

450. 

12 

1900. 

Page  93. 

Art.  304. 

1. 

$143. 

0 

$384. 

3. 

$65,875. 

4. 

$113.75. 

5. 

$291.83i. 

6. 

$133.31^^. 

7. 

$247.60. 

8. 

$63.05. 

9. 

$1.2U. 

10. 

$3.70. 

11. 

$67.71^. 

12. 

$10.50. 

13. 

$841.15. 

u. 

$1156.97. 

Page  94. 

Art.  306. 

1. 

$75. 

2 

$358.75. 

3. 

$22.13. 

4. 

$53.50. 

5. 

$1125. 

6. 

$612. 

7. 

$3281.fi5. 

8. 

$382. 

9. 

$24.06. 

IG. 

$45.56. 

11. 

$34. 

12. 

$1567.50. 

13. 

$187.50. 

14. 

$281.25. 

15. 

$125.25 

IG. 

$370. 

17. 

$2750. 

IS. 

$46.25 

19. 

$50.25 

20. 

$86.25. 

21. 

$156.25. 

aftei 


26. 


O, 


280 


Art.  307. 

2. 

$2507.96. 

Page  95. 

2. 

$3324.46. 

3. 

$4321.26. 

4. 

$5282.20. 

5. 

$8096.48. 

0. 

$3364.72. 

Page  96. 

Art.  309. 

1. 

231b. 

n 

4627.5  yd. 

.'. 

61b. 

■'{. 

763.2  yd. 

5. 

876.5  doz. 

6. 

371  lb. 

7_ 

81.5  yd. 

S. 

115.2  acres. 

9. 

689  yd. 

JO. 

123  lb. 

11. 

$40.96. 

Page  97. 

Art.  311. 

1. 

$21.91. 

r9 

$92.68. 

.?. 

$70.07. 

4. 

$28.35. 

5. 

$148.28. 

6. 

$31.68. 

7. 

$25.31. 

8. 

$89.10. 

9. 

$63.63. 

10. 

$35.65. 

Art.  313. 

1. 

$24.75. 

2. 

$14:66. 

3. 

$158.76. 

4. 

$123.18. 

5. 

$53. 

6. 

$38.75. 

7. 

$19.80. 

8. 

$100.32. 

0. 

$140.81. 

10. 

$34.78. 

11. 

$68296.35. 

12. 

$2819.31. 

Page  98. 

Art.  315. 

1. 

$4.02. 

ANSWERS. 

(0 

$4.71. 

S. 

$66.21. 

3. 

$78.70. 

4. 

$13.84. 

Page  105. 

5. 

$2014.46. 

Art.  328. 

6. 

$2016.46. 

1. 

$943.54. 

7_ 

$941.63. 

o 

$57269.94. 

8. 

$21.06. 

0. 

$2.77. 

Page  106. 

10. 

$19.51. 

Art.  329. 

11. 

$296.24. 

1. 

$115.68. 

12. 
13. 

$15.77. 
$56.72. 

2. 

$560.50. 
$1528.75. 

14. 

$66162.39. 

4. 

$190.33. 

Page  99. 

Art.  317. 

J. 

$272.35. 
$429.14. 

1. 

$28.56. 

Page  107. 

a 

$28.08. 

7. 

$524.03. 

o. 

$52.83. 

s. 

$1718.01. 

4. 

$39.29. 

9. 

$168.68. 

5. 
6. 

'  $24.66. 
$27.15. 

10. 

$322.29. 

S. 

$55.67. 
$58.33. 

Page  112. 

0. 

$174.78. 

Art.  355. 

10. 

$132.36. 

1. 

450  min.  51  sec. 

11. 

$121.68. 

■) 

23  hr.  5  min.  29 

12. 

$242.79. 

sec. 

13. 

$333.31. 

3. 

7  da.    1   hr.   30 

u. 

$109.57. 

min.  51  sec. 

15. 

$198.91. 

4- 

24  yr.  Imo.  5  da. 

16. 

$28.16. 

6  min. 

17. 

$11.57. 

5. 

63321  hr. 

IS. 

$12.06. 

0. 

Jan."l4,  1889. 

19. 

$60.89. 

7. 

3405  da. 

20. 

$30.14. 

8. 

1169  da. 

21. 

$15.62. 

9. 

11  mo.  9  da. 

.->-> 

$1941.92. 
Page  101. 

10. 

1  yr.  10  mo.  19 
da.  19^  hr. 

Art.  326. 

Page  113. 

1. 

$9.03. 

11. 

No  difference. 

Page  102. 

12. 

939613.7  sec. 

2. 

$49954.08. 

Art.  358. 

3. 

$191.52. 

1. 

35°  54'. 

2 

24'  16' 46". 

Page  103. 

3. 

3S.  r  49'4r. 

4- 

$1812.31. 

4. 

1296000. 

Art.  327. 

5. 

4907'. 

1. 

$46731.53. 

G. 

205737. 

7. 

136°  2'. 

Page  104. 

8. 

21600'. 

2 

$967.31. 

9. 

8°  39'. 

Page  115. 

Art.  363. 

1.  2  hr.  30  min.  24 

sec. 

2.  41  min.  40  sec. 

3.  57  min.  44  sec. 

after  3  a.  m. 

4.  1  hr.  52  min.  8 

sec. 

5.  16   min.   past    8 

p.  m. 

.    Art.  365. 

1.  47°  59'. 

2.  35°  13'  E. 
S.     74°  58'. 

Page  116. 

4.  36°  52' 20"  N. 

5.  77°  1'. 

Art.  366. 

1.  180°. 

2.  180°. 

3.  4  5  16  p.  m. 

4.  6  33  40  a.  m. 

6    min.    of    1 

p.  m. 

6  a.  m. 

10  9  20  p.  m. 

1  21  2U  p.  m_ 

3  min.  48  sec. 

past  7  a.  m. 

Page  117. 

Art.  371. 

1.  £256  4  s. 

-'.  248  s.  3  d. 

3.  £54  6  s.  10  d. 

4.  £195  4  s.  4  d.  t 

far. 

Alt.  373. 

1.  6480  d. 

2.  956  far. 

3.  38853  d. 

4.  39450  far. 

5.  13206  far. 

Page  118. 

Art.  375. 

1.  $350.26. 

2.  4525.80. 

3.  $63544.40. 


ANSWEKS. 


281 


4.     $15.03. 
r,.     154.29. 

Art.  3  77. 

1.  £38  3  d.  2.4  far. 

2.  £63  7  8.  10  d. 

3.  £513  14  s.  3  d. 

3  far. 

4.  £751  14  s.  3  d. 
r,.     £32621  2  s.  8  d. 

1  far. 

Page  124. 

Art.  393. 

1.  7653  pwt. 

2.  155948  gr. 

3.  4  lb.  11  gr. 

If.     5  lb.  3  oz.  2  pwt. 
9gr. 

5.  432  gr. 
6'.     -i^  pwt. 

9.     1  OZ.  13  pwt.  18 

gr- 
it;.    12  pwt.  12  gr. 

11.     AVlb. 

12.  mib. 

13.     7    oz.    14     pwt. 

4.8  gr. 
U.     18  pwt.  2.4  gr. 

15.  .297616  + lb. 

16.  .875  oz. 

17.  472  lb.  1  oz.  12 

pwt.  8  gr. 

Page  125. 

IS.     211  lb.  11  oz.  19 
pwt.  21  gr. 

19.  2  pwt.  20.  4  gr. 

20.  .0067+. 

21.  81b.  2  oz.  13  pwt. 

6gr. 

22.  18  lb.  9   oz.   14 

pwt.  2  gr. 

23.  $8032.50. 

24.  $1924.39. 

25.  5  oz.  2  pwt.  17 

gr- 

26.  1  oz.   13  pwt.   8 

gr. 

27.  $11655. 


28.  73  lb.  2  oz.  8  pwt. 

19  gr. 

29.  $154.22. 

30.  $360.67  gain. 

Page  127. 

Art.  395. 

1.     34669  lb. 

..'.     15  T.  12  cwt.  75 

lb. 
.i.     12  cwt.  50  lb. 

4.  56  lb.  4  oz. 

5.  7  cwt.  68   lb.   6 

.4  oz. 

6.  12  cwt   50  lb. 

"-  7089    T 

8.  llS-cwt. 

9.  .24125  cwt. 
10.     .99996875  T. 

Page  128. 

n.     30  T.  1  cwt.  94 
lb.  11  oz. 

12.  $72.81. 

13.  2  T.  5  cwt.  84  lb. 

Art.  397. 

1.  17  lb.  9  oz.  5  dr. 

1  sc. 

2.  5896  dr. 

.'f.     11  oz.  3  dr.  2  sc. 
.8gr. 

5.  63  sc. 

6.  6  lb.  9  oz.  6  dr. 

11.5  gr. 
;.     1  lb.  2  oz.  4  dr. 

1  sc.  4  gr. 
S.     7  lb.  3  oz.  4  dr. 

12  gr. 
9.     6  lb.  2  oz.  6  dr. 

1  sc.  8  gr. 

10.  15  lb.   10  oz.  5 

dr.  2  sc.  8  gr. 
;/.     11  oz.  2  sc.  4^i 
gr- 

Page  129. 

Art.  398. 

1.  $1477.27. 

2.  $2255.25. 

3.  61b.  lOoz.  15gr. 

4.  18  lb.  6.43?  oz. 


$254.49. 
$5.04. 


Art.  400. 

1.  928  pt. 

2.  599  pt. 

3.  180bu.  3qt.2pt. 

4.  7  qt.  If  pt. 

Page  130. 

.7.     5  bu.  1  pk.  1  qt. 

1  pt. 
',.     83  bu.  1  pk.  3qt. 

IfPt. 
;.    $184.12. 

Art.  403. 

1.     5932  gi. 

,?.     31  bbl.  7  gal.  1 

pt.  3gi. 
V?.     651.168  gi. 

4.  6  gal.  2  qt.  1.16 

g'- 
,J.     $71.75. 
6.     7  gal.  2  qt.  1  pt. 

V^  gi- 
;.     13  gal.  1  pt.  1  gi. 

5.  $19.16. 

'.).     $72.58,  gain. 

Page  131. 

Art.  403. 

1.  1579A  pt. 

2.  134.5 +  pt.  gain. 

3.  $.52  gain. 

4.  $5.25,  gain. 

5.  $7.99  less. 

Page  132. 

Art.  409. 

J.     127002  in. 

2.  39  mi.  155  rd.  4 

3'd.  3  in. 

3.  T^tyY"?- 

/,.     213  rd.  1yd.  2  ft. 

6  in. 
o.     %\  rd. 

6.  173  rd.  2  yd.  1ft. 

812  in. 

7.  .892+. 

S.  123  mi.  162  rd.  3 
yd.  1  ft.  4  in. 

9.  1593  mi.  312  rd. 
2  yd.  1  ft.  8  in. 


10.     484  mi.  53  rd,  1 
yd.  2  ft.  6  in. 

Page  134. 

Art.  424. 

1.  35676648  sq.  in. 

2.  1344984421  sq.ft. 

3.  112  A.  40sq.  rd. 

261  sq.  ft.  51.84 
sq.  in. 

Page  135. 

4.  110  sq.  rd. 

-5-      44aOS440g  sq.  mi. 

6.  .9382+ A. 

7.  101  sq.  rd.  2  sq. 

yd.  21.6  sq.  in. 

8.  4  A.  83  sq.  rd.  6 

sq.  yd.  64|  sq. 
in. 

9.  31  i  squares. 

10.  63  yd. 

11.  90  ft. 

12.  230  ft. 

13.  25|  A. 

U.     $107156.25. 
ir,.     $3277.97. 

16.  357i  rd. 

17.  414?  ft. 

18.  130i  A. 

19.  Not  any. 

20.  2  sq.  rd. 

21.  2%%. 

22.  4||  A. 

23.  58^  rd. 

24.  12. 

25.  26A.llsq.  rd.  4 

sq.  yd.  5  sq.  ft. 
36  sq.  in. 

26.  $60.06. 

27.  $10.56. 

28.  9  A.  110  sq.  yd. 

3  sq.  ft.  54  sq. 
in. 

29.  38  A.  59  sq.  rd. 

12  sq.  yd.  5  sq. 
ft.  112  sq.  in. 

30.  640  rd. 

31.  320  rd. 

Page  136. 

32.  60  yd. 

33.  $40.30. 

34.  $58594.44. 


282 


ANSWERS. 


S5.     $188404. 

36.  $15.63. 

37.  43  rolls. 

3S.  128J-  yd.,  130 
yd.,  and  $327. 
25. 

39.  486U|. 

40.  12  sq.  ft. 

41.  $9.46. 
4^.     28512. 

43.  52.177+  ft.  and 

104.354+  ft. 

44.  147840. 

45.  $74.36. 

46.  $204.73. 

Page  137. 

Art.  430. 

1.  1. 

2.  9. 

3.  16. 
^.  25. 
5,  81. 
6\  100. 
7.  9801. 
-s'.  65536. 

Page  139. 

Alt.  439. 

1.  14. 

S.  15. 

^.  12. 

4.  24. 

5.  35. 
6-.  75. 
7.  206. 
<9..  11.2. 
.0.  7.09. 

10.     21.954. 
i/.     5.07. 


Page  140 

12. 

10.3156. 

13. 

f. 

14. 

f. 

l.'>. 

.968  H». 

10. 

.85+. 

17. 

5510.8 +. 

18. 

68548.66+. 

Art.  446. 

1. 

10. 

.<> 

49.777. 

5. 
6. 

3. 

9. 
10. 
11. 
i:. 

13. 

U. 

15. 


Page  141. 

108.25+. 
60.81  +. 
56.796;  113.592. 
208.71  +  ft. 
1866.76+ ft. 
660  ft. 
124.03  + ft. 
77.88  + ft. 
72.56  + ft. 
452    id.,    8 

10.92  + in. 
226.42 +  rd. 
720  rd. 
208.80  rd. 


ft. 


Page  142. 

Alt.  448. 

1.     211618.48  in. 
S.     1  mi.  68  elf.  1  rd. 
161. 
3rd.  181.59.4in. 
7624  1. 

7ch.l41.  2.27f  iu. 
8927,\  ft. 
1924*  rd. 
S.     16  ch.  91  1.  5.22 

in. 
.'/.     2738  steps,    llf 
in.  rem. 

Page  145. 

Art.  459. 

1.  10  cu.  yd.   1533 

cu.  in. 

2.  6178581. 

3.  8cu.  yd.972cu. 

in. 

4.  7200  cu.  ft. 

5.  508i  cu.  yd. 
247  j\  pch. 
Trr/sTff  cu.  yd. 

cV.     .2615  +  cu.  yd. 
9.     ^3^  cu.  ft. 

10.  14  cu.  ft.,  302.4 

cu.  in. 

11.  $2320.76. 

12.  $2656.78. 

13.  650471  bricks. 

14.  47i  cd. 

15.  39^ j  cd. 

16.  26  ft.  9/5  in. 

17.  $227.11. 

18.  54468f  lb. 


19. 

20. 
■21. 


S. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 


1:. 
IS. 
19. 


4531i  lb. 

46656. 

10  cu.  yd.  20  cu. 

ft.  1339icu.  in. 
659.709+  pch.; 

254013  bricks; 

$1169.41. 

Page  146. 

198cu.  ft. 
27.52+ bu. 

Page  150. 

Art.  474. 

12. 

25. 
48. 
404. 

12.898  + 
49.21+. 
36.1 +. 


.8+. 

.92+. 

2.90 +. 

15.177+. 

.160+. 

.97+. 

Page  151. 

10  ft.  1  +  in. 
47  ft.  5  + in. 
5ft.  4  +  in.wide; 

10    ft.    9  + in. 

high,  and  37  ft. 

7  +  in.  long. 
14  ft.  2  + in. 
7  ft.  7  + in.  deep, 

and  15  ft.  2  + 

in.  square. 

Page  153. 

Art.  492. 

78  sq.  ft. 
85.498  sq.  yd. 
32|  A. 
360  A. 
259.182  ft. 
472. 68  + ft. 
12.27  + A. 
88. 6  + ft. 
29.7+ in  square. 


lu.     $37.70. 

11.     26649.9  gal. 

/.-.     45  sq.  ft. 

Page  154. 

13.  26|  sq.  yd. 

14-  201.0624  sq.  in. 

15.  33.5104  cu.  ft. 

16.  1260. 9 +. 
1:.  4564.2  + mi. 
IS.  117.6264  gal. 
19.  22.3074  gal. 

Page  157. 

Art.  502. 

1.  £359,  12  s.  1.8. 

+  far. 

-'.  .Tan.  3,  1870. 

3.  420  sq.  ft. 

4.  2.295+ A. 

5.  122.18  + pch. 

6.  17280  shingles. 

7.  27^\  rd. 
<s'.  $65.66+. 
9.  $14.04. 

10.  117.77  + A. 

11.  220.85  qt. 

12.  17  da.  13  hr.  13 

min. 

13.  143823.788  fr. 

14.  1424} « I  fr. 

15.  3647641 » A  bbl. 
ir,.     58  min.,  39.2  + 

sec.    after     10 
o'clock  A.  M. 
n.     $16.56. 

18.  222.63+. 

19.  2471 5.7+. 

20.  15yr.  80da.  5hr. 

33  min.  20  sec. 

21.  $11.20. 

Page  158. 

22.  65iyd. 

23.  28T.  8.9  +  cwt. 

24.  42"  30'. 

25.  229.2784  lb. 

20.  14  min.  20  sec. 
after  1  p.  M. 
Aug.  19,  1909. 

27.  10710.38 +  fr. 

28.  $145.10. 

29.  5199.306  + mar. 

30.  72-r\  ft. 


ANSWERS. 


283 


34. 


So. 
36. 


39. 
40. 
41. 
^?. 
43. 


44. 
45. 

46. 

47. 

48. 
4-9. 
■50. 
■61. 

52. 

.53. 
54. 


1. 
2. 

4. 
5. 


20  A.  4  sq.  rd. 

31  sq.  ft. 
7  Mm.  6  Km.  7 

Hm.   2  Dm.  6 

M.  8  dm.  .438 

cm. 
Coat  $7.54,  and 

vest  $3.28. 
78  sq.    rd.,    106 

sq.     ft.,     25.56. 

sq.  in. 
$25.03  gain. 
$414.95. 
302.379  sq.  ft. 
$4026.10. 
$446511.63. 
G17A.,120sq.rd. 
$.31. 
$890.25. 
25.19  + ft. 

Page  159. 

$6189.51. 

161|  ft. 

$188.67. 

$6431.23. 

64  rd.  9.6+ ft. 

$14.39. 

2.886+. 

5  da. 

24  lb.    10  oz.  7 

pwt.,  2.95  gr. 
14  ft.  4  + in. 
24.33  da. 
2  mi.  108  rd.  G 

ft. 


Page  1G2. 

Art.  533. 

1.  5. 

2.  72. 

3.  120. 

4.  100. 

5.  300. 
0-.  ■^. 

7.  360. 

8.  36. 

9.  340. 


Art.  684. 

150  A. 
872  sheep. 
$28653.75. 
198.24  lb. 
2  1b.  10  pwt. 


6.  $621. 

7.  $86.79  gain. 

8.  $14375  and  $8125. 

9.  $32.50. 

10.  725|  bu. 

Page  163. 

11.  $17500. 

12.  $682.50. 

13.  $266666.661. 

14.  $22720.50. 

15.  m  A. 

16.  $ll04. 

17.  $36453.10. 

18.  $42.50  gain. 

Art.  .526. 

14100. 
21500. 
10860. 

4.  657. 

5.  20000. 
6 

Page  164. 

7.  $4000. 

8.  250  A. 
.'/.     $108. 

Art.  537. 

1.  400  bales. 

2.  $740.88. 

3.  $1475. 

4.  $693. 33i. 

5.  $40300. 

6.  $538294.12. 

7.  $90000. 
S.  101040. 
.''.  70000. 

10.     $180000. 

Page  165. 

Art.  529. 

1.  20j?. 
25;?. 
3.     50^. 

4.  m%. 

33i^. 

s.    20oor^'. 

9.  3000;^. 

Art.  630. 

1.  25%. 

2.  Z\%. 


33i^. 
'2.%. 


53ijr. 

3623\^. 

33i,'?. 

56}?;. 

25;?. 

Page  166. 

Art.  .->33. 

1.10,    ami.    per. 

cent. 
1.75,    ami.    per. 

cent. 
2.10,    amt.    per. 

cent. 
1.161,  amt.  per. 

cent. 
1.87^,  amt.  per. 

cent. 

Art.  533. 

1.05  per  cent. 
1.09,',;?. 

i.40r;. 

Art.  535. 

.85,  difference 

per  cent. 

.62^,  difference 

per  cent. 

.99^,  difference 

per  cent. 

.96  J,  difference 

per  cent. 

.30,  difference 

per  cent. 

Art.   536. 

712';. 
68|;f. 

.60,      difference 
per  cent. 

Page  167. 

Art.  538. 

1650. 

1695. 

462. 

1180. 

277.2. 

2580. 

840. 


8.  637. 

9.  450. 


10. 


Art.  539. 

$14512.50. 

$456. 

$11200. 

816. 

3537. 

$886.50. 

Art.  541. 
12. 
150. 
945. 


Page  168. 

4. 

612. 

5. 

1200. 

6. 

500. 

7. 

5. 

8. 

567  ft. 

Art.  543. 

1. 

$2843.75. 

2. 

581  i  A. 

3. 

$2053.13. 

4. 

$632.50. 

Art.  544. 

1. 

600. 

2. 

400. 

■'■ 

300. 

Page  169. 

4. 

100. 

5. 

$4000. 

Art.  545. 

$1000. 
$4.51. 
$7950. 
$480. 
$106. 
$3200. 
$30000. 
$10000. 
500  pupils. 
Invest,  in  farm. 
$280. 

Page  170. 

Art.  648. 

AAS4r 
0*25  ,. 

68  lb. 


284 


ANSWERS. 


5.  80^. 

4.  123333333. 33i. 

5      23H.    425,    and 
33J. 

6.  77  yr. 

7.  $108. 

5.  H.|200,3I.$170 

C.  $15. 
9.     $1081.25. 

10.  25  lb.  warp,  and 

71i  lb.  rags. 

11.  $160,  and  $224. 

12.  $113.78. 
IS.     \\%  gain. 

Page  171. 

1^.     $-i{387.50. 

15.     $1045.45  +. 

IQ.     216:;+. 

11.     21^Y<,  21M!f.  ""J 

ij?.  100  bead. 

19.  &^^. 

^0.  25^. 

21.  $6344.40. 

S2.         ISi'Tr. 

23.  $1000  loss. 

24.  400. 
f5.  3000. 
26.  77|i|^. 
i-?.  10  yd. 
£8.  $4856.25. 
29.  $60250. 

Page  172. 

50.  Grazing,  504  A. ; 

grain,  420  A  ; 
timber,  936  A. 

51.  $192  C. 

32.  A  $93840.  and  B 

$69360 

33.  $22400. 

34.  Not  any. 

35.  7500,  9750,  6825, 

and  9555. 

36.  Clover,  450;  tim- 

othy, 450;  or- 
chard grass,  150 
and  50  red  top. 

j:.  $81.20.  $101.50, 
$182.70. 

38.     16|?. 


39.  $22629.31. 

40.  $1750,  $3062.50, 

$6125,  and 
$8575. 

41.  25600  T. 

42.  Wife,      $21750; 

D.,  $10000;  Y. 
S., $12500;  and 
E.  S.,  $13750. 

Page  173, 

Art.  557. 

1.  $9. 

2.  $48. 
J.     $750. 

Page  174. 

$50. 

$32. 

$320. 

$225. 

$2100. 

$700. 

.\Tt.  558. 

1.  $592.50  gain. 

2.  $3997.50  loss. 
J.     $677.25  loss. 

4.  $9.47  gain. 

5.  $184.92  gain. 

6.  $6  gain. 

:.     $10  16  lo-ss. 
S.     $22.97. 
;/.     $133.59. 

Page  175. 

Art.  560. 

1.  $100. 

„'.  $3500. 

J.  $10000. 

4.  $4400. 
o.  $40. 
C.  $300. 
?.  $900. 

5.  $1050. 
5.  $1. 

Art.  561. 

;.  $57.. 50. 

.?.  $500  and  $625. 

J.  $600. 

.^.  $700. 

5.  $7085.71. 

tJ.  200  A. 


r.     $2750. 

.V.     $240. 
[>.     $5000. 

Page  176. 

Art.  563. 

1.  m. 

2.  5^. 

3.  m^. 

4.  20^. 

5.     50;?. 

<;.   33i:?.    ^ 

7.     150^. 
9.     20f^. 

Art.  564. 

1.     150.?. 
•?      25^. 

Oats.     \%i%. 

33i'?. 

9,<?  gain. 

771'?  gain. 

Page  177. 

20;,  profit. 
131;;  gain. 
IH'T. 

663f;. 

6i  loss. 
5O5?  gain. 

m- 

$23163.12. 

Art.  566. 

1.  $100. 

2.  $40. 

3.  $15. 
^.  $75. 
5.  $1000. 
/;.  $25600. 

Art.  567. 

1.  $365.71. 

2.  $442.50. 

3.  $7500. 

Page  178. 

4.  $7501. 

5.  1080  lb. 
';.     $4000. 

Art.  569. 

1.  $153.60. 

2.  $500. 


3.  48  yr. 

4.  75^    nitre,   12i^ 

sulphur,     and 
12i^  charcoal. 

Page  179. 

5.  44. 

6.  $11.25. 

7.  $1260  and  $840. 

8.  $2865. 

9.  $2890.80 

10.    374?. 

ii.  $800. 

12.  $2. 

iJ.  $3111.11. 

i.^  33^^;. 

15.  2200  bbl. 

id.  461';. 

17.  $31i. 

i^.  No  gain  or  loss. 

19.  40,?. 

20.  $5. 
~'i.  33i?. 
£f.  $74.25. 

23.  $1972.50. 

24.  100?. 

Page  180. 

25.  $9350. 
fg.  27^'. 

27.  $200  and  $2.50, 

28.  16f  loss. 
f9.  $2700. 
30.  $288. 
.?i.  $10000. 

:?f .  $4:312.50  gain.  . 

S3.  $110. 

34.  31?  loss. 

.55.  $16.80. 

36.  94*?. 

57.  $155. 

55.  $4. 

39.  M(t. 

40.  $5.2o  loss. 

41.  $59320. 

Page  181. 

42.  $2.62*.  6i?,  and 

$175  loss. 

43.  25?. 

^^.  15<?- 

45.  $420. 

^6.  18^?  loss. 

47.  83i?. 


ANSWERS. 

285 

J^. 

$3. 

Art.  579. 

3. 

U%.                          ' 

Page  200. 

49. 

684^. 

1.  m. 

4. 

$10635.53. 

Art.  63  7. 

50. 

$5000  cost,  and 

2.     39.4375;^. 

5. 

33i^. 

1. 

$5645.50. 

$12441.60. 

3.        51yV5^. 

I!. 

2000    lb.       and 

^ 

$1744.80. 

SI. 

$450. 

4-   eeisj^. 

$16.20  com. 

3. 

$1178.80. 

52. 

$6050.85. 

S.     Tf^. 

7. 

$1818.60. 

.',. 

$2188.60. 

53. 

$160. 

6'.     37.791$«'. 

S. 

$1184.60. 

5. 

$10656.40. 

54. 

$125. 

7.     1A$?. 

9. 

$1097.40. 

Art.  O.tS. 

55. 
56. 

lAf^  loss. 
$58.60. 

Page  188. 

Art.  588. 

10. 

UJ. 

Page  195. 

1. 

0 

$904.70. 
$686.35. 

Page  182. 

J.     $211. 

11. 

$1888. 

3. 

$900.90. 

57. 

78.596  yd. $19.40 

$483.96. 

12. 

194122.3  lb.  and 

4. 

$3565.68. 

$757.20. 

$10531.70. 

58. 

gain. 

9U  gal- 
$1100onionsand 

Page  180. 

13. 

$2773.17. 
1890    bbl.     and 

5. 

59. 

Art.  590. 

$1.11. 

^• 

$1970.90. 

$750  potatoes. 

1.     $36.09. 

14. 

$648. 

S. 

$3812.50. 

60. 

9^. 

2.     $368.48. 

ir,. 

$1508.57      com. 

Page  201. 

61. 

$681.82  pear. 

Page  190. 

and   $16091.43 

Art.  649. 

and   $1071.43 

Art.  592. 

m. 

5if 

1. 

$91.20. 
Page  202. 

62. 

apple. 
$2273.40     gain. 

1.     $64.50. 

17. 
13. 

10666^  yd. 
$7021.21. 

63. 

and  12Hi^. 
$454.59. 

Page  192. 

Art.  606. 

19. 

20. 

45.278?.'. 
$30859.74. 

■> 

$386.25. 

64. 

$4000. 

J.     $180. 

2.  $75. 

3.  $250. 

4.  $275. 

5.  $277.38. 

21. 

253     bbl.,     and 

Art.  651. 

65. 

100500  corn,  and 

$9.68. 

1. 

1;;,  $62.50. 

75375  wheat. 

2. 

$3550. 

66. 

C,  $27.18J;     H, 

Page  196. 

3. 

$537.50. 

$326.25;      and 

/■O 

$31403.75. 

4. 

4^  mills,  $70. 

S,  $1.81+. 

6.     1375.90. 

23. 

26250  corn,  and 

''. 

$35.35. 

19200  barley. 

Page  184. 

Page  193. 

24. 

5^. 

Page  203. 

Art.  575. 

Art.  608, 

25. 

$5044.29. 

0. 

3^    mills,    $888 

1. 

2. 

$24.48. 
$.^)1.30. 

7.     $5000. 
J.     $24500. 

26. 

Remitted  $566^, 
com.       $113Jt, 

83.13,     S.     T. 
$27553.77. 

3. 

4- 
5. 
6. 

$660. 
$5670. 
$19. 13  gain. 
200  yd. 

■J.     $282. 

4.  $8672. 

5.  9000  bu. 
0".     50  bales. 

28. 
29. 

and  rate  16i^L 
Loss  of  $280.26. 
$26023.50. 
21071.52  bu. 

*8. 
9. 

2  J      mills;      24 

mills. 
4i  mills; $110.63 
$54.05. 

''• 

$1515.64. 

30. 

Barley,      14333^ 

10. 

$462  33. 

S. 

A,     $72.50. 

Art.  610. 

bu. ;  hops,  560- 

11. 

$.005ff,and$85.- 

Paj;*'  185. 

J.     $12140. 
-'.     30000  lb. 

67.13  lb.;    and 
com.  $508.40. 

IL 

17. 
1.25669  /. 

Art.  .'>~~. 

$30. 
$30. 

3.     145  doz. 

1. 
2. 

Page  194. 

Page  199. 

Page  206. 

0. 

$12571.43. 

4.     $400.80. 

Art.  6.14. 

.Art.  6  7  7. 

4. 

$9. 

5.     1500     A.,     and 

1. 

$22.80. 

1. 

$5000. 

5. 

$60. 

$202.50. 

r> 

$300.38. 

•:■> 

470. 

6. 

25f.'. 

r,.     17501b. 

3. 

$8000. 

3. 

$1600,       $2400, 

7. 

2ff/. 

4- 

$72. 

$2000. 

Art.  611. 

.'). 

$234. 

4- 

$1576.50. 

Page  18«. 

J.     $07.50. 

6. 

40  gal. 

5. 

$523.75. 

S. 

$1000. 

J.     225  bbl. 

1      ''• 

$144. 

6. 

$4450. 

286 

Page  207. 

22. 

$2161.54. 

ERS. 

Page  218. 

50. 

$.53. 

7.     $27411.17. 

25. 

$4981.67. 

Art.   715. 

Art.  716. 

8.     $114.54.55. 

2i. 

$44.75. 

1. 

$4.38. 

1. 

$79.20. 

9.     $14791.67. 

25. 

$780. 

2. 

$5.25. 

9 

$29.96. 

10.     $16853.56. 

26. 

$100.83. 

3. 

$3.71. 

3. 

$80.84. 

11.  ir<- 

27. 

$96.75. 

4. 

$17.44. 

4- 

$104.45. 

12.     $1967.96. 

28. 

$16975. 

5. 

$9.24. 

13.     $60. 

Page  213. 

6. 

$1.13. 

Page  220. 

U.     $6093.40. 
15.     $3200. 
le.     $18242.66. 

29. 
SO. 

$74.94. 
$2707.18. 

S. 

0. 

$5.83. 
$3.33. 
$1.76. 

5. 

$73.41. 
Art.  718. 

n.     H,       $26522.73; 

SI. 

$4310.74. 

1". 

$5.70. 

1. 

$5.18. 

M.    $43099.43; 

32. 

$2040.15. 

11. 

$2.34. 

2 

$11.53. 

A,     $23207.39; 

33. 

$2766.55. 
$1969.62. 
$3519.75. 

12. 

$8.02. 

3. 

$3.62. 

Phoenix,  $265- 

34. 

13. 

$6.32. 

4. 

$2.80. 

22 .73  and  Prov- 

35. 

14- 

$3.00. 

5. 

$5.19. 

ident,  $26522.73 

36. 

$2837.92. 

15. 

$24.58. 

6. 

$16.81. 

IS.     $5000. 

Page  214. 

16. 

$1.15. 

7_ 

$3.95. 

19.     $47500.  and  $3- 

Art.   705. 

17. 

$4.37. 

8. 

$5.10. 

9375.                    ' 

1. 

$343.75. 

18. 

$11  65. 

9. 

$16.95. 

1 
Page  208. 

20.     G.,     $630;     H., 
$150;  and  M., 

2. 
3. 
4. 

$4099.71. 

$337.41. 

$857.01. 

$27500. 

$826.23. 

19. 

20. 

$1.34. 
$10.96. 

Page  219. 

10. 
1. 

$4.  .50. 

Art.   719. 

$191.26. 

$337.50. 

6. 

21. 

$6.61. 

2 

$90.88. 

$142.50  gain. 

22 

$19.71. 

.n\% 

Page  215. 

23. 

$25.80. 

Page  221. 

Art.   707. 

24. 

$10.60. 

3. 

.$45.89. 

Page  211. 

1. 

$445.94. 

25. 

$25.38. 

Art.   :03. 

2. 

$10344.83. 

26. 

$2.39. 

Page  222. 

1.     $258.30. 

3. 

$600. 

27. 

$4.86. 

.\rt.  729. 

2.  $47.67. 

3.  $75.60. 

4. 
5. 

$1000.      ' 
$291.85. 

28. 

29. 

$8.22. 
$6:i8. 

1. 
.9 

$448.70. 

$1546.70. 

$366.60. 

$2422.30. 

$12726.80. 

4.     $364.50. 

6. 

$739.13. 

30. 

$2.82. 

5.     $457.10. 

Page  212. 

7. 
1. 

$1954.63. 
Art.   709. 

31. 
32. 
S3. 

$7.79. 
$1.79. 

$8.78. 

3. 

4. 
5. 

6.  $131.39. 

7.  $675.13. 

8.  $570. 

0 
3. 

7?. 
6<?. 

lOr;. 

34. 
S5. 
36. 

$14.30 
$1.41. 
$1.47. 

Page  223. 

Art.   734. 

9.     $322.58. 

37. 

$.42. 

1. 

372.96. 

10.  $523.80. 

11.  $102.10. 

12.  $223.12. 

6. 

7. 

38. 
39. 

40. 

$12.87. 
$10.50. 
$7.70. 

2. 
3. 
4. 

$96.45. 
$459.34. 

$417.84. 

13.     $2800.53. 

Page  216. 

41. 

$.12. 

5. 

$1198.09. 

H.     *  74.43. 

Art.   711. 

■i'- 

$.35. 

6. 

$2319.22. 

15.     $300.28. 

1. 

2yr.  5  mo.  24  da. 

43. 

$1.29. 

IG.     $220.26. 

■■> 

3  yr.  10  mo.  12 

44. 

$.70. 

Page  226. 

n.     $1662.50. 

da. 

45. 

$5.50. 

.\rt.   737. 

IS.     $132.86. 

3. 

April  25.  1881. 

40. 

.S16.53. 

1. 

*27>t>.75. 

19.     $1798.30. 

4. 

11  mo. 

47. 

$11.66 

2 

$1020.30. 

20.     $717.27. 

5. 

Sep.  22,  1889. 

4S. 

$5.70. 

S. 

H^. 

21.     $791.78. 

G. 

12  yr.  6  mo. 

40. 

$29.17 

4. 

S%. 

ANSWERS. 

28? 

Art.   738. 

Page  230. 

9.     Jan.     28,     1889; 

Page     248. 

1.     $889.58. 

Art.   744. 

Term  of  Dis., 

5. 

Nov.  15,  1888. 

2.     $1773.73. 
Page  227. 

1. 

$540. 

$99.75,  and  $38.- 

27  days;  Pro., 
$384.71. 
10.     Feb.    39,    1888; 

6. 

7. 
3. 

Dec.  29,  1887. 
Jan.  14,  1889. 
Jan.  11,  1888. 

S.     $1618.33. 

<0. 

19 da.,  $799.09. 

4.     $6386.77. 

3. 

$1654.61. 

11.     Aug.  6,  1888;  66 

Page  249. 

5.     $879.71. 

* 

days;  $664.99. 

.Art.  803. 

Page  231. 

12.     Mar.  3, 1889;  178 

1. 

Nov.  12,  1888. 

Art.   730. 

1.  4  yr.  2  mo. 

2.  6.716J?. 

4- 
5. 

Interest,  $12.29. 
$.55,    better    to 

days;  $3383.44. 
13.     Dr.,  $125.39. 

3. 
4. 

Dec.  21,  1888. 
Jan.  10,  1889. 
Feb.  13,  1889. 

3.     X3.23:?. 

pay  cash. 

5. 

June  13,  1888. 

A.     $920.08. 

6. 

No  difference. 

Page  238. 

6. 

May  9,  1889. 

5.     $20.72. 

7. 

Loss  $11.82. 

14.     $1900.41  to  their 

7. 

May  14,  1888. 

6.     $8681.12. 

8. 

$1533.15. 

credit. 

8. 

Mar.  7,  1889. 

7.  m. 

8.  3  yr.   7  mo.    24 

0. 
10. 

$7.48. 
A'i'o  loss. 

15.     $3865.30. 

Page  255. 

da. 

11. 

$7481.30. 

Art.  780. 

Art.  806. 

9.     $10505.94. 

12. 

Cashoffer,$8.37. 

1.     $330. 

1. 

Nov.  28,  1886. 

10.     7i  yr. 

13. 

^ll'/o  profit. 

2.     $1350. 

.? 

Feb.     11,    1887, 

11.     $6856.53. 

u. 

Guin  .$1303. 

3.     $933.87. 

$300. 

12.     ^\\i. 

15. 

$44.85,and8|i';. 

4.     $3461.96. 

3. 

$313.18. 

13.     $4644.61. 

IC. 

$683.33. 

5.     $3150. 

4. 

601.73. 

17. 

$4000. 

';.     $691.13. 

5. 

Oct.  14,  1888. 

Page  228. 

IS. 

$340.13. 

7.     $175.08. 

Page  256. 

U.     May  18,  '89. 

If: 

$39.79. 

G. 

$100.     Nov.  35, 

15.     Gain  $875. 

Page  240. 

1890. 

16.  $2728.82. 

17.  Grace,  $7678.96; 

Page  232. 

Alt.  790. 

/.     $199.37. 

7. 
S. 

Sep.  21,  1889. 
Dec.  33,  1886. 

Mabel.  $7031.- 
85 ;          Flora, 
$4333.02. 
18.     $299.20. 

20. 
21. 

23. 

$37.73. 
$1756.27. 
$372.58. 
$9736.94. 

2.  $533.68. 

3.  $1100.85. 
4-     $4.07 

9. 
10. 

Sept.  4,  1887. 
Jan.    24,    1888. 
$348.88. 

19.  m',L 

20.  $1373.81. 

24. 

$10855.79. 

Page  241. 

11. 

Page  257. 

July    24,     1887, 

21.  $4128.37. 

22.  725  M. 

Page  236. 

J.     $100.53. 
'-■.     $1550.07. 

12. 

$431. 
$300,    Dec.    27, 

23.     $2660. 

Art.  778. 

7.     $1.73. 

1887,    $300.43. 

24.     50  years. 

1. 

Bk.  Dis.,  $9.38; 

S.     $1890.50. 

13. 

Feb.  16,  1889. 

25.     Herbert,  $5938.- 

Pro.,  $740.63. 

14. 

$100,  May  17, '89 

66;    Theodore, 

2 

Bk.  Dis.,  $1.67; 

Page  242. 

$98.93. 

$4847.73. 

Pro.,  $284.83. 

Art.  793. 

3. 

Bk.  Dis.. $23. 08; 

1.     $203.98. 

Page  258. 

Page  229. 

Pro.,  $1303.93. 

..'.     $640.87. 

15. 

$400,  Jan.  4, '88; 

2Q.     $536.95,  better  to 

,;.     $39.18. 

$435.23. 

invest  in  land. 

27.     $33884.38. 

Page  237. 

4.     $563.53. 

IG. 

Dec.  7,  1888. 
Page  261. 

28.     $1508.75. 

4- 

Bk.  Dis.,  $4.47. 

Page  247. 

Art.  820. 

29.     $13006.80. 

5. 

Proc,  $988.37. 

Art.  801. 

1. 

4. 

SO.     Chas.,  $5364.99; 

G. 

Gain,  $11.35. 

1.    Oct.  16,  1888. 

2. 

28, 

John, $4590.03; 

7. 

Bk.  Dis., $13.71; 

2.     Oct.  13,  1888. 

3. 

7. 

Walter,  $3895.- 

Pro.,  $1251.71. 

,;.     Sep.  7,  1888. 

4. 

81. 

34. 

S. 

Proc,  $1749.47. 

4.     Mar.  30,  1889. 

5. 

$288. 

288 


ANSWERS. 


6.  $4.30+. 

7.  733i  ft. 
S.  478  bu. 
9.     $2812.50. 

10.     2  yr.  9  mo.  22i 
days. 

Page  262. 

Art.  823. 

J.     26JA. 

350  rd. 

$567. 

$1290. 

$384.75. 

124^  yd. 

74^. 
S.    93  da. 

Page  265. 

Art.  841. 

1.  A..  $26250;  and 
B.,  $15750. 

£.  $6589.41,  gain; 
Hadley,$2758.- 
36;  and  Hunt, 
$3831.05. 

S.  Whole  capital, 
$58800;  D's 
gain,  $1200. 

4.  A,   $9409.-52;  B, 

$7939.29:    and 
C,  $7351.19. 

5.  N.Insol.,$4543.- 

75.   N.  In  vest., 
$2321.25. 

6.  Gained,  $12090; 
A'sP.W.,$504- 
5;B'sP.W.,$37- 
95;  Solv.,$8840. 

7.  Harrison,  $7000; 

Morton,  $1000. 

Page  267. 

.Vrt.   842. 

7.     A,     $270.92;  B, 

$346.15;  C, 

$276.93. 

i".     A,    $596.53;  B, 

$559  25  ;  C. 

$994.22. 


3.  Martin's  gain, 
$1737.93;  Eaton's 
gain,  $1862.07; 
Martin's  P.  W., 
$3737.93;  Eaton's 
P.  W.,  $6362.07. 

4.  A,     $17.40;     B, 

$29.70;  andC, 
$48. 

5.  A's  investment, 
$8491.30;  B's  in- 
vestment, $6065.- 
22;  C's  invest- 
ment. $4043.48. 

6.  A,  $4548.39;  B, 

$2951.61.  I 

7.  A.  $36;B,  $32;C, 
$20;    and  D,  $4. 

S.     A,  $731.57;    B, 

$1483.93;    and 

C,  $784.50. 

9.  Net  gain,  $5200; 

Olsen's   share  of 

net  gain,  $1640.- 

27;     Thompson's 

share  of  net  gain, 

$3559.73. 

Page  268. 

10.  Simmons,  $170- 

20.59,  and  Saw- 
yer, $16329.41. 

11.  Drews'  gain, 
$8058.14;  Allen's 
gain,  $6837.21  ; 
Bracketl's  gain, 
$6104.65. 

12.  B,  $838.10;  and 

A,  $1561.90. 

13.  Martin,    $3126.- 

53;  Gould, 

$3104.46;    and 
Cole,  $1269.01. 

Page  269. 

2.  Hopkins's  gain, 
$4873.27  ;     Haw- 


ley's  gain,  $4526.- 
73. 

3.  Charles,  12f: 
.John,  8j^;  Walter, 

Page  270. 

4.  Net  resources, 
$23564.25  ;  net 
solvency,  $23,564.- 
25;  net  gain, 
$1064.25. 

5.  Investm'tof  each, 
$9866? :  Gray's 
gain.  $2161.69  ; 
Snyder's  gain, 
$1894.93;  Dillon's 
gain,  $2843.38  ; 
Dillon's  P.  W., 
$16210.04;  Sny- 
der's P.  W.,  $72- 
61.60 ;  Gray's  P. 
W.,  $11028.35. 

6-.  Phelps,  $8000 
Rogers,  $18300 
Wilder,  $30900, 

7.  Smith,  $16170.43 
Jones,  $11990.32 
Brown,  $9339.25. 
Smith's  P.  W.,  $2- 
8870.43  ;  Jones's 
P.W.,  $16990.32; 
Brown's  P.  W., 
$16239.25.      i 

S.  Burke, $17533.33; 
Brace,  $17833.33; 
Baldwin, $17633.- 
33. 

9.  Loss,  $22747.09. 
Briggs'  loss,  $5- 
907.62;     Parson's 

•  loss,  $16839.47; 
net  insolvency, 
$2187.09;  Briggs' 
P.  W.,  $127.38; 
Parson's  insolv., 
$2314.47. 


1". 


11. 


I  l-'. 


13. 


14- 


Page  271. 

Bradley's  gain, 
$7517.61  ;Maben'8 
gain.  $8362.39 ; 
Bradley's  P.  W., 
$23417.61 ;  Ma- 
ben's  P.  W.,  $26- 
362.39. 

Wilkins'  share, 
$1000;  Ames' 
share,  $3000. 
$1744.5.24.  A's  P. 
W.$30165.84;B's 
P.  W.  $22417.09; 
C's  P.  W.  $22417.- 
08.  A's  share  of 
gain,  $6033.17  -, 
B's  share  of  gain, 
$4483.42 ;  C's 
share  of  gain, 
$4483.41. 
A's  capital,  $386- 
5.80;  B's  capi- 
tal, $2577.20;  in- 
solvency, $5557  ; 
A's  insolvency, 
$3334.20;  B's  in- 
solvency, $2222.- 
80. 


Page  272. 

Net  resources, 
$94735;  net  gain, 
$44335;  Clay's 
share  of  gain, 
$17932.89;  Hard's 
share  of  gain,  $2-i- 
380.43;  Dunn's 
share  of  gain,  $2- 
021.68.  Clay's  P. 
W.  at  closing,  $3- 
4132.89 ;  Hard's 
P.  W.  at  clos- 
ing, $53080.43  ; 
Dunn's  P.  W.  at 
closing,  $7521.68. 


] 


D    000  878  084    3 


^^T, 


>5Jr,- 


